The fitting and modeling of skewed, complex, symmetric, and asymmetric datasets is an exciting research topic in many fields of applied sciences, notably lifetime, medical, and financial sciences. The Heavy tailed Nadarajah Haghighi model is introduced by compounding the heavy tailed family and Nadarajah Haghighi distribution in this paper. The so obtained model has three parameters accounting for the scale and shape of the distribution. The proposed distribution's fundamental characteristics such as probability density, cumulative distribution, hazard rate, and survival functions are provided, and several key statistical properties are established, and several entropy information measures are proposed. The estimation of model parameter is performed via the ML procedure. Further, different simulation experiments are conducted to demonstrate the performance of proposed estimator’s using some measure, like the average estimate, the average bias, and associated mean square error. Finally, we are applied our proposed model to analyze three different real datasets. In our illustration, we are compared the practicality of the recommended model with several well-known competing models.

In this paper, we derive the distribution of eigenvalues for a 2×2 random confusion matrix stemming from a machine learning (ML) evaluation problem. Additionally, we present the distributions of both the matrix’s trace and the difference between the two traces of random confusion matrices. We also illustrate the application of these derived distributions in calculating the superiority probability of four baseline ML models.

Floods are a common natural disaster whose severity in terms of duration, water resource volume, peak and accumulated rainfall-based damage are likely to differ significantly for different geographical regions. In this paper, we first propose a novel hourly flood index (SWRI24−hr−S), derived from normalising the existing 24-hourly water resources index (WRI24−hr−S) in literature, to monitor flash flood risks on an hourly scale. The proposed SWRI24−hr−S is adopted to identify a flood situation and derive its characteristics, such as the duration (D), volume (V) and peak (Q). The comprehensive result analysis establishes the practical utility of SWRI24−hr−S in identifying flood situations at seven study sites in Fiji between 2014 and 2018 and deriving their characteristics (i.e., D, V, and Q). Secondly, this study develops a vine copula-probabilistic risk analysis system that models the joint distribution of flood characteristics (i.e., D, V, and Q) to extract their joint exceedance probability for the seven study sites in Fiji, enabling probabilistic flood risk assessment. The vine copula approach, particularly suited to Fiji’s study sites, introduces a novel probabilistic framework for flood risk assessment. The results show moderate differences in the spatial patterns of joint exceedance probability of flood characteristics in different combination scenarios generated by the proposed vine copula approach. In the worst-case scenario, the probability of any flood event occurring where the flood volume, peak and duration are likely to exceed the 95th-quantile value (representing an extreme flood event) is found to be less than 5% for all study sites. The proposed hourly flood index and the vine copula approach can be feasible and cost-effective tools for flood risk monitoring and assessment. The methodologies proposed in this study can be applied to other data-scarce regions where only rainfall data is available, offering crucial information for flood risk monitoring, assessment, and the development of effective mitigation strategies.

This paper concentrates on the general birth-death processes with two different types of catastrophes. The Laplace transform of transition probability function for birth-death processes with two-type catastrophes are is successfully expressed with the Laplace transform of transition probability function of the birth-death processes without catastrophe. The first effective catastrophe occurrence time is considered. The Laplace transform of its probability density function, expectation and variance are obtained.

The main source of electricity worldwide stems from fossil fuels, contributing to air pollution, global warming, and associated adverse effects. This study explores wind energy as a potential alternative. Nevertheless, the variable nature of wind introduces uncertainty in its reliability. Thus, it is necessary to identify an appropriate machine learning model capable of reliably forecasting wind speed under various environmental conditions. This research compares the effectiveness of artificial neural networks (ANN) and convolutional neural networks (CNN) in predicting wind speed across three locations in South Africa, characterised by different weather patterns. Empirical results show that CNN outperforms ANN in accurately forecasting wind speed under different weather conditions. This superiority is likely due to the inherent architectural attributes of CNNs, including feature extraction capabilities, spatial hierarchy learning, and resilience to spatial variability. These results could be useful to decision-makers in the energy sector.

Estimating parameters for multiple datasets can be time consuming, especially when the number of datasets is large. One solution is to sample from multiple datasets simultaneously using Bayesian methods such as adaptive multiple importance sampling (AMIS). Here we use the AMIS approach to fit a von Misses distribution to multiple datasets for wind trajectories derived from a Lagrangian Particle Dispersion Model driven from a 3D meteorological data. The primary objective is to characterise the uncertainties in wind trajectories in a form that can be used as inputs for predictive models of wind-dispersed insect pests and pathogens of agricultural crops for use in evaluating risk and in planning mitigation actions. Our results show that AMIS can significantly improve the efficiency of parameter inference for multiple datasets.

The Mann-Whitney effect is a measure for comparing survival distributions between two groups. The Mann-Whitney effect is interpreted as the probability that a randomly selected subject in a group survives longer than a randomly selected subject in the other group. Under the independence assumption of two groups, the Mann-Whitney effect can be expressed as the traditional integral formula of survival functions. However, when the survival times in two groups are not independent each other, the traditional expression of the Mann-Whitney effect has to be modified. In this article, we propose a copula-based approach to compute the Mann-Witney effect with parametric survival models under dependence of two groups, which may arise in the potential outcome framework. In addition, we develop a Shiny web app that can implement the proposed method via simple commands (https://nkosuke.shinyapps.io/shiny_survival/). Through a simulation study, we show the correctness of the proposed calculator. We apply the proposed methods to two real datasets.

Energy distance is used to eigenvalue distributions in this new Wishart distribution study, providing a new perspective on high-dimensional data. Multivariate statistics relies on Wishart distribution to explain covariance matrices. This analysis shows how degrees of freedom and covariance matrix configurations affect eigenvalue dispersion. We use energy distance based on eigenvalue distributions to assess Wishart distribution differences without the "curse of dimensionality" via extensive simulation simulations. This approach improves measurement accuracy and processing efficiency in distribution comparisons and enables new analytical pathways in scientific fields that need data covariance comprehension. Our results show that degrees of freedom and matrix structure considerably affect energy distances, expanding the theoretical framework and practical applications of statistical models.

Asymptotic theories for fractional cointegrations have been extensively studied in the context of time series data, with numerous empirical studies and tests developed. However, most of the previously developed testing procedures for fractional cointegration were primarily designed for time series data. This paper proposes a generalized residual-based test for fractionally cointegrated panels with fixed effects. The test development is based on bivariate panel series with the regressor assumed to be fixed across cross-sectional units. The proposed test procedure accommodates any integration order between [0,1], and it is asymptotically normal under the null hypothesis. Monte Carlo experiments demonstrate that the test exhibits better size and power compared to a similar residual-based test across varying sample sizes.

The future index prices are viewed as a critical issue for any trader and investor. For this craving, various models have shown up in the literature. The Geometric Brownian motion (GBM) is one of the popular models. This work examines four types of GBM as per the presence of memory or kind of volatility. These models include classical GBM with memoryless and constant volatility assumptions, SVGBM with memoryless and stochastic volatility assumption, GFBM with memory and constant volatility assumption, and SVGFBM with memory and stochastic volatility assumption. These models are utilized in an empirical study to forecast the future index price of Energy Sector in the Saudi Stock Exchange Market. The assessment was led by utilizing two error standards, mean square error (MSE) and mean absolute percentage error (MAPE). The outcomes showed that the SVGFBM occupies the highest level of accuracy according to smallest values of MSE and MAPE. While the accuracy of GBM come in the tail of the list models under study. These results have affirmed the positive affection of combining memory and stochastic volatility assumption into the GBM model, which agreed findings of numerous earlier works. Furthermore, the findings showed the GFBM models are more accurate than GBM models regardless of the type of volatility. While, under the same type of memory, the models with stochastic volatility assumption are more accurate than the corresponding models of constant volatility assumption. In general, all models considered in this work uncovered high accuracy through the value of MAPE ≤ 10%. This indicates of the ability of applying these models in a real financial environment. Through this empirical study, we can say that the Energy Sector in Saudi Arabia is predictable and stable and afterward we urge financial investors and stockholder to trade and invest in this sector.

The two-parameter distribution introduced by Kumaraswamy (1980) is a very flexible alternative to the Beta distribution with the same (0,1) support. Originally proposed in the field of hydrology, it has subsequently received a good deal of positive attention in both the theoretical and applied statistics literatures. Interestingly, the problem of testing formally for the appropriateness of the Kumaraswamy distribution appears to have received little or no attention to date. To fill this gap, in this paper we apply the “biased transformation” methodology proposed by Raschke (2009) to several standard goodness-of-fit tests based on the empirical distribution function. A simulation study reveals that these (modified) tests perform well in the context of the Kumaraswamy distribution, in terms of both low size distortion, and respectable power. In particular, the “biased transformation” Anderson-Darling test dominates the other tests that are considered.

IMPORTANCE: On December 10, 2021, the FDA published a Determination Letter, along with a Statistical Review and Evaluation Report, and concluded that under the non-informative prior, the local Bayesian optimal interval design (BOIN) design, in its revised form, can be designated fit-for-purpose for identifying the maximum tolerated dose of a new drug, assuming that dose-toxicity relationship is monotonically increasing. Although setting the BOIN design parameter p.tox = 1.4 * target.DLT.rate is recommended in almost all BOIN methodology articles and is the default value in the R package BOIN, it's unclear if the choice of p.tox should only depend on the target DLT rate and if certain range of p.tox could produce the same BOIN boundary table. DESIGN: In this simulation study, following parameters were varied one at a time, using R package BOIN, to explore their effects on the equivalence intervals of p.saf and p.tox: 1) target DLT rate, 2) n.earlystop, 3) cutoff.eli, 4) cohortsize, and 5) ncohort. And a simple 3+3 design was used as an example to explore equivalent sets of BOIN design parameters that can generate the same boundary table. RESULTS: When the early stopping parameter n.earlystop is relatively small or the cohortsize value is not optimized via simulation, it might be better to use p.tox < 1.4 * target.DLT.rate, or try out different cohort sizes, or increase n.earlystop, whichever is both feasible and provides better operating characteristics. This is because if the cohortsize was not optimized via simulation, even when n.earlystop = 12 and cohortsize > 3, the BOIN escalation/de-escalation rules generated using p.tox = 1.4 * target.DLT.rate could be exactly the same as those calculated using p.tox > 3 * target.DLT.rate, which might not be acceptable for some pediatric trials targeting 10% DLT rate. The traditional 3+3 design stops the dose finding process when 3 patients have been treated at the current dose level, 0 DLT has been observed, and the next higher dose has already been eliminated. If additional 3 patients were required to be treated at the current dose in the situation described above, the decision rules of this commonly used 3+3 design could be generated using BOIN design with target DLT rates ranging from 18% to 29%, p.saf ranging from 8% to 26%, and different p.tox values ranging from 39% to 99%. These BOIN parameters also need to satisfy a set of conditions. Meaning: This study demonstrates the importance of interpreting BOIN design parameter p.tox as an interval of toxicity rates that are considered too toxic, rather than one prespecified value that corresponds to the lowest toxicity probability that is deemed overly toxic. When designing a dose-finding trial using BOIN, it is important to perform simulation studies to identify equivalent sets of BOIN design parameters that can generate the same boundary table so that we can better compare the safety properties of different boundary tables.

This study introduces a novel computational approach for testing conditional independence (BB CI test) within causal Directed Acyclic Graphs (DAGs), leveraging Bayesian non-parametric bootstrap and machine learning techniques. Our method offers an alternative for validating the assumptions underpinning causal DAGs. Through simulation studies and an industrial case analysis, we demonstrate the test procedure in accurately assessing conditional independence, comparing it with the Generalized Covariance Measure (GCM) test. Our findings suggest that the BB CI test is advantageous in scenarios where existing methods may falter due to violations of model assumptions. This research contributes to the causal inference literature by providing a computational tool for researchers and practitioners to validate causal models.

Partial Information Decompositions (PIDs) aim to categorize how a set of source variables provide information about a target variable redundantly, uniquely, or synergetically. The original proposal for such an analysis used a lattice-based approach and gained significant attention. However, finding a suitable underlying decomposition measure is still an open research question, even at an arbitrary number of discrete random variables. This work proposes a solution to this case with a non-negative PID that satisfies an inclusion-exclusion relation for any f-information measure. The decomposition is constructed from a pointwise perspective of the target variable to take advantage of the equivalence between the Blackwell and zonogon order in this setting. We prove that the decomposition satisfies the axioms of the original decomposition framework and guarantees non-negative partial information results. We highlight that our decomposition behaves differently depending on the used information measure, which can be utilized for different applications. We additionally show how our proposal can be used to obtain a non-negative decomposition of Rényi-information at a transformed inclusion-exclusion relation, and for tracing partial information flows through Markov chains.

The only cases where exact distributions of estimates are known is for samples from exponential families, and then only for special functions of the parameters. So statistical inference was traditionally based on the asymptotic normality of estimates. To improve on this we need the {\it Edgeworth expansion} for the distribution of the standardized estimate. This is an expansion in $n^{-1/2}$ about the normal distribution, where $n$ is typically the sample size. The 1st few terms of this expansion were originally given for the special case of a sample mean. In earlier work we derived it for {\it any } standard estimate, hugely expanding its application. We call an estimate $\hat{w}$ of an unknown vector $w\in R^p$, a {\it standard estimate}, if $E\ \hat{w}\rightarrow w$ as $n\rightarrow \infty$, and for $r\geq 1$ the $r$th order cumulants of $\hat{w}$ have magnitude $n^{1-r}$ and can be expanded in $n^{-1}.$ Here we give another huge extension. We give the expansion of the distribution of {\it any smooth function} of $\hat{w}$, say $t(\hat{w})\in R^q,$ giving its distribution to $n^{-5/2}$. We do this by showing that $t(\hat{w})$, is a standard estimate of $t(w)$. This provides far more accurate approximations for the distribution of $t(\hat{w})$ than its asymptotic normality. %NOT USED: The building blocks of the Edgeworth expansions are the {\it cumulant coefficients } of the estimate. These cumulant coefficients are also needed for bias reduction, Bayes estimates and confidence regions. We give {\it chain rules} for the cumulant coefficients of $t(\hat{w})$ in terms of those of $\hat{w}$ and the derivatives of $t(w)$, up to those needed for 5th order Edgeworth expansions for $t(\hat{w})$ and its {\it tilted expansion}, useful for the tail of the distribution.

A model of a single-server queuing-inventory system (QIS) with catastrophes and a finite waiting room for consumer customers (c-customers) has been developed. When a catastrophe occurs, all inventory in the system's warehouse is destroyed, but c-customers in the system are still waiting for restocking. In addition to c-customers, negative customers (n-customers) are also taken into account, which displaces one c-customers (if any). The policy (?, ?) is used to replenish stocks. If upon arrival of the c-customer the inventory level is zero, then according to Bernoulli’s scheme this c-customer is either lost or joining the queue. The mathematical model of the studied QIS is constructed in the form of a two-dimensional continuous-time Markov chain (2D CTMC). Exact and approximate methods for calculating the steady-state probabilities of constructed 2D CTMCs are developed and closed-form expressions are derived for calculating the performance measures. The results of numerical experiments are presented, demonstrating the high accuracy of the developed approximate formulas, as well as the behavior of performance indicators depending on the reorder point. In addition, an optimization problem is solved to obtain the optimal reorder point to minimize the expected total cost.

Considering that cryptocurrencies are now present in practically every financial transaction because they are widely accepted as an alternate means of making payments and exchanging currencies, academics and economists have more opportunities to study cryptocurrency prices. Over the years, investors, traders and investment banks have found it difficult to predict the closing daily price of Ethereum due to its rapid price fluctuation. The daily closing price of cryptocurrency is essential to consider when trading or investing in Ethereum. This report focuses on carrying out a comparative study of the predictive capabilities of deep machine learning algorithms with a stacking ensemble modelling framework using daily historical observations of the price of Ethereum obtained from Coindesk, tweets extracted from Twitter ranging from the 1st of August 2022 to the 8th of August 2022 and other five covariates (closing price lag1, closing price lag2, noltrend, daytype and month) engineered from the closing price of Ethereum. Seven models are used to compute the forecasts for the daily closing price of Ethereum; these are the recurrent neural network, ensemble stacked recurrent neural network, gradient boosting machine, generalized linear model, distributed random forest, deep neural networks and stacked ensemble for gradient boosting machine, generalized linear model, distributed random forest and deep neural networks. The main evaluation metric used is the mean absolute error. According to MAE, RNN forecasts outperform the other model’s forecasts in this study, producing an MAE of 0.0309.

This paper provides the first-time ever analytic modelling for Pointwise Stationary Fluid Flow Approximation(PSFFA) model of the non-stationary

The SK model proposed by Sherrington and Kirkpatrick is an important mean field model for spin glasses. The dependence structure of Hamiltonians of the SK model is considered in this paper. We have investigated the covariance matrix of Hamiltonians of the SK model, and establish a relationship between the rank of the corresponding covariance matrix and the dimension of the space spanned by the Hamiltonians. Finally, we also give an upper bound for the rank of the covariance matrix, which might be the best.

The paper explores the Kullback-Leibler divergence formalism (KLDF) applied to the stable MG1 queue manifold. More potentially, both service time probability and cumulative functions which make KLDF exact are obtained. The credibility of KLDF is justified through consistency axioms. Some potential applications of Kullback-Leibler divergence to Biometry are presented. The paper concludes with closing remarks combined with some challenging open problems and the next phase of research.

This paper provides a new discovery that adds to the contemporary theory of modelling of non-stationary queues, by capturing analytically the behavior of the root parameter, for the G1/M/1 PSFFA model of non-stationary Ek/M/1 queue. Additionally, it has been also revealed that the resulting root parameter reduces to the PSFFA model of non-stationary queue. Potential applications of PSFFA to Computer engineering are provided. Some challenging open problems are addressed combined with closing remarks and future research directions.

In descriptive statistics, U -statistics arise naturally in producing minimum-variance unbiased estimators. In 1984, Serfling considered the distribution formed by evaluating the kernel of the U -statistics and proposed generalized L-statistics which includes Hodges-Lehamnn estimator and Bickel-Lehmann spread as special cases. However, the structures of the kernel distributions remain unclear. In 1954, Hodges and Lehmann demonstrated that if X and Y are independently sampled from the same unimodal distribution, X − Y will exhibit symmetrical unimodality with its peak centered at zero. Building upon this foundational work, the current study delves into the structure of the kernel distribution. It is shown that the kth central moment kernel distributions (k > 2) derived from a unimodal distribution exhibit location invariance and is also nearly unimodal with the mode and median close to zero. This article provides an approach to study the general structure of kernel distributions.

Descriptive statistics for parametric models are currently highly sensative to departures, gross errors, and/or random errors. Here, leveraging the structures of parametric distributions and their central moment kernel distributions, a class of estimators, consistent simultanously for both a semiparametric distribution and a distinct parametric distribution, is proposed. These efficient estimators are robust to both gross errors and departures from parametric assumptions, making them ideal for estimating the mean and central moments of common unimodal distributions. This article also illuminates the understanding of the common nature of probability distributions and the measures of them.

This paper investigates the search for an exact analytic solution to a temporal first-order differential equation that represents the number of customers in a non-stationary or time-varying

We introduce a novel stochastically correlated two-factor (i.e., bivariate) diffusion process under the square-root format, for which we obtained, analytically, the corresponding solutions for the conditional moment generating functions and conditional characteristic functions. Such solutions recover verbatim those of the uncorrelated case. We believe that the The aforementioned model encompasses a range of processes similar to that produced by a bivariate square-root process in which entries are correlated in the standard way, i.e., via a constant correlation coefficient. Note that, for the latter, closed-form solutions for the conditional characteristic functions are not available. We focus on the financial scenario of obtaining closed-form expressions for the exact price of a zero-coupon bond and Asian option prices via a Fourier cosine series method.

The current study characterizes the transient M/M/1 queue manifold info-geometrically, through devising Fisher Information matrix(FIM) and its inverse(IFIM). Additionally, the impact of stability on the existence of IFIM and explore the geodesic equations of motion has been revealed. More potentially, the paper discusses the relationship between stability and the Gaussian curvature, as well as the connections between queueing theory, information geometry, , Riemannian geometry, and the Theory of Relativity.

The paper explores the Kullback-Leibler divergence formalism (KLDF) applied to the stable MG1 queue manifold. It explores the analytic forms of state probabilities and their maximization based on entropy functionals, subject to normalization and mean value constraints. The credibility of KLDF is justified through consistency axioms, and the application of Rényi's and Tsallis's formalisms on a stable M/G/1 queue is examined, resulting in novel state probabilities and insights into information theory of Queue Learning.

Population events such as natural disasters, pandemics, extreme weather, and wars might cause jumps that have an immediate impact on mortality rates. The recent COVID-19 pandemic has demonstrated that these events should not be treated as nonrepetitive exogenous interventions. Therefore, mortality models incorporating jump effects are particularly important to capture the adverse mortality shocks. The mortality models with jumps, proposed by Cox. et al. , Chen and Cox , and Özen and Şahin , which we consider in this study, differ in terms of the duration of the jumps - transitory or permanent, frequency of the jumps, and size of the jumps. To illustrate the effect of jumps, we also consider benchmark mortality models without jump effects, such as the Lee-Carter , Renshaw and Haberman and the Cairns-Blake-Dowd models. We discuss the performance of all the models by analysing their ability to capture the mortality deterioration caused by COVID-19. We use data from different countries to simulate the mortality rates for the pandemic years and examine their accuracy in forecasting the mortality jumps due to the pandemic. Moreover, we also analyse the impact of mortality jump models on catastrophe bond pricing.

The theory of Ismail's non-extensive maximum entropy solution (NME) is described in detail. It is used as an inductive inference technique for heavy-tailed queues with a non-robust mean queue length and a non-extensive "long-range" interaction. In our novel method, we substitute the non-robust mean queue length for the conventional Pollaczeck-Khinchin mean queue length. In other words, the new non-extensivity parameter q will be included in the resulting state probability function. Numerical portraits are provided to capture the influential effect of the derived formalism, , on the stable M//1 queue with heavy tails. More potentially, some applications of Hurst Exponent to social computing and connected health are provided. Conclusion with some challenging open problems and possible future research pathways are given.

The current paper provides a cutting-edge information data length- theoretic approach to the transient M/M/Infinity queueing system. This is a first time ever investigation that unifies information data length and queueing theories. The significant impact of both time and the number of states for the transient M/M/ Infinity queue on both the upper and lower bounds of the obtained information data length is observed and noted. The paper concludes with some challenging open problems, combined with concluding remarks and future research pathways.

From a differential geometric perspective, information geometry (IG) aims to characterise the structure of statistical geodesic models. The research done for this paper offers a novel method for modelling the information geometry of a queuing system. From the perspective of IG, the manifold of the transient M/M/infinity queue is described in this context. The Fisher Information matrix (FIM) as well as the inverse of (FIM), (IFIM) of transient M/M/infinity QM are devised. In addition to that, new results that uncovered the significant impact of stability of QM on the existence of (IFIM) are obtained. Moreover, the Geodesic equations of motion of the coordinates of the underlying transient are obtained. Also, it is revealed that stable QM is developable (i.e., has a zero Gaussian curvature) and has a non-zero Ricci Curvature Tensor (RCT). Novel stability dynamics of transient M/M/iInfinity queue manifold is revealed by discovering the mutual dual impact between the behaviour of (RCT) and the stability(instability) phases of M/M/Infinity QM. Furthermore, a new discovery that presents the significant impact of stability of queue manifold and the continuity of the unique representation between M/G/1 queue manifold and Ricci Curvature Tensor (RCT). The information matrix exponential (IME) is devised. Also, it is shown that stability of the devised (IME) enforces the instability of M/M/ Infinity QM. More interestingly, novel relativistic info-geometric queueing theoretic links are revealed .

The Rényi entropy functional was optimised for the novel findings of this investigation under various conditions. Their research's findings suggest that the complete spectrum of Lévy stable distributions is covered by generalised t-distributions. Additionally, an exposition was undertaken to prove that the Lévy distribution generalizes Tsallisian distribution, rather than the reverse. The current study is a strong generalization of an existing research work in the literature.

An exposition is undertaken to reveal the significant impact of the time-dependent root parameter, on the stability of Pointwise Stationary Fluid Flow Approximation (PSFFA) model of the non-stationary queueing system. This opens new grounds to stability analysis of non-stationary queueing systems. Potential PSFFA applications of IoT are highlighted.

Information geometry is a mathematical framework that analyses the structure of statistical models using concepts from differential geometry. It treats families of probability distributions as manifolds, where the parameters of each model determine the coordinate charts. By applying info-geometric tools, we can gain insights into the characteristics of these models. The approach involves characterizing the queueing system's manifold using information geometry and presenting the exponential of the information matrix. This integration of information geometry with queueing theory provides a novel perspective for analyzing the dynamics of queueing systems, incorporating relativistic and Riemannian concepts. Some applications to E-health are highlighted. Finally, closing remarks and the next phase of research.

The partial information decomposition (PID) framework is concerned with decomposing the information that a set of (two or more) random variables (the sources) has about another variable (the target) into three types of information: unique, redundant, and synergistic. Classical information theory alone does not provide a unique way to decompose information in this manner and additional assumptions have to be made. One often overlooked way to do this decomposition is using a so-called measure of union information – which quantifies the information that is present in at least one of the sources – from which a synergy measure stems. In this paper, we introduce a new measure of union information based on adopting a communication channel perspective, compare it with existing measures, and study some of its properties. We also include a comprehensive critical review of characterizations of union information and synergy measures that have been proposed in the literature.

Ridge regression is one of the most popular shrinkage estimation methods for linear models. Ridge regression effectively estimates regression coefficients in the presence of high-dimensional regressors. Recently, a generalized ridge estimator was suggested by generalizing the uniform shrinkage of ridge regression to the non-uniform shrinkage, which was shown to perform well under sparse and high-dimensional linear models. In this paper, we introduce our newly developed R package “g.ridge” (the first version published on 2023-12-07 at https://cran.r-project.org/package=g.ridge) that implements both the ridge estimator and generalized ridge estimators. The package equips with the generalized cross-validation for automatic estimation of shrinkage parameters. The package also includes a convenient tool for generating a design matrix. By simulations, we test the performance of the R package under sparse and high-dimensional settings with the normal and skew-normal error distributions. From the simulation results, we conclude that the generalized ridge estimator is superior to the benchmark ridge estimator based on “glmnet”, and hence, it can be the most recommended estimator under sparse and high-dimensional models. We demonstrate the package using the intracerebral hemorrhage data.

Multiple imputation with chained equations (MICE) is a widespread approach to account for missing data in empirical research. Combining MICE with bootstrapping, that is, repeatedly resampling with replacement from the data to estimate variances and confidence intervals of statistics of interest, is not straightforward. The current document provides an overview of how to use bootstrapping with imputed data in Stata. Two main approaches (impute first and then bootstrap or vice-versa) are discussed and shortly compared. Code is provided for Stata.

In the literature, estimation of weighted extropy are least considered. In this paper some non parametric estimators of weighted extropy are given. The validation and comparison of the estimators are implemented with the help of simulation study and data illustration. The usefulness of the estimators are demonstrated using real data sets.

Intrusion detection is generally carried out by matching network traffic patterns with known attack patterns or by identifying abnormal network traffic patterns. One statistical methodological approach used in intrusion detection is Statistical Process Control (SPC) by constructing a control chart. Hotelling’s T2 control chart is a multivariate control chart commonly used to monitor the mean process. The performance of the T2 chart in monitoring mean shifts can be increased if a robust estimator is utilized. Based on previous research, T2 based on the Fast-MCD estimator has good performance in monitoring low to medium outlier contaminated data. Therefore, the MRCD estimator can be used to detect intrusion. On the other hand, this research focuses on developing a bootstrap-based robust Hotelling’s T2 charts with Fast-MCD and MRCD estimators for evaluating performance in detecting intrusion on intrusion detection datasets. Based application of UNSW-NB15, the proposed chart has better performance than the conventional T2 and Fast-MCD-based T2 despite the longer execution time.

In this paper, we shall introduce the interval-valued spatial error model. Based on the idea of least square method of single-valued case, we give the parameter estimator for interval-valued spatial error model. The theoretical properties of the proposed estimator are proved. Finally, we give the numerical analysis and a real example.

In today’s competitive landscape, achieving customer-centricity is paramount for the sustainable growth and success of organisations. This research is dedicated to understanding customer preferences in the context of the Internet of Things (IoT) and employs a two-part modeling ap-proach tailored in this digital era. In the first phase, we leverage the power of the Self-Organizing Map (SOM) algorithm to segment IoT customers based on their connected device usage patterns. This segmentation approach reveals three distinct customer clusters, with the second cluster demonstrating the highest propensity for IoT device adoption and usage. In the second phase, we introduce a robust Decision Tree methodology designed to prioritize various factors influencing customer satisfaction in the IoT ecosystem. We employ the Classification and Regression Tree (CART) technique to analyze 17 key questions that assess the significance of factors impacting IoT device purchase decisions. By aligning these factors with the identified IoT customer clusters, we gain profound insights into customer behaviour and preferences in the rapidly evolving world of connected devices. This comprehensive analysis delves into the factors contributing to customer retention in the IoT space, with a strong emphasis on crafting logical marketing strategies, en-hancing customer satisfaction, and fostering customer loyalty in the digital realm. Our research methodology involves surveys and questionnaires distributed to 207 IoT users, categorizing them into three distinct IoT customer groups. Leveraging analytical statistical methods, regression analysis, and IoT-specific tools and software, this study rigorously evaluate the factors influencing IoT device purchases. Importantly, this approach not only effectively clusters the IoT Customer Relationship Management (IoT-CRM) dataset but also provides valuable visualizations that are essential for understanding the complex dynamics of the IoT customer landscape. Our findings underscore the critical role of logical marketing strategies, customer satisfaction, and customer loyalty in enhancing customer retention in the IoT era. This research makes a significant contri-bution to businesses seeking to optimize their IoT -CRM strategies and capitalize on the oppor-tunities presented by the IoT ecosystem.

We present a formalism to assess the safety of the COVID19 vaccines and infer a causal relation between vaccine administration and adverse effects, in particular death. We use data on adverse effects from VAERS covering the time interval [01-10-2020, 31-12-2021] (downloaded on 1 Feb 2022). We measure a vaccine fatality rate of order 0.01 for all vaccine manufacturers and identify a strong heterogeneity in the vaccine toxicity across vaccine lots, spanning up to four orders of magnitude. We compute the correlation between vaccination dates and death dates, and produce an estimate of the time lag that maximises the correlation, finding that the onset of adverse effects happened statistically (i.e. 32-41 percentiles) within one day after the vaccine administration and that death happened statistically (i.e. 22-50 percentiles) within about two weeks after the vaccine administration. These results provide data-based insights that can guide public health measures and health insurance policies.

Genome-wide association studies (GWAS) have successfully revealed many disease-associated genetic variants. For a case-control study, the adequate power of an association test can be achieved with a large sample size, although genotyping large samples is expensive. A cost‐effective strategy to boost power is to integrate external control samples with publicly available genotyped data. However, the naïve integration of external controls may inflate the type I error rates if ignoring the systematic differences (batch effect) between studies, such as the differences in sequencing platforms, genotype calling procedures, population stratification, and so forth. To account for the batch effect, we propose an approach by integrating External Controls into the Association Test by Regression Calibration (iECAT-RC) in case-control association studies. Extensive simulation studies show that iECAT-RC not only can control type I error rates but also can boost statistical power in all models. We also apply iECAT-RC to the UK Biobank data for M72 Fibroblastic disorders by considering genotype calling as the batch effect. Four SNPs associated with Fibroblastic disorders have been detected by iECAT-RC and the other two comparison methods. However, our method has a higher probability of identifying these significant SNPs in the scenario of an unbalanced case-control association study.

In the paper we present a new test for comparison of the means of multivariate samples with unknown distributions. The test is based on the comparison of the distributions of the distances between the samples’ elements and their means using univariate two-sample Kolmogorov-Smirnov test. The activity of the suggested method is illustrated by numerical analysis of the real-world and simulated data.

Spatial regression models have garnered significant attention across several disciplines, including functional magnetic resonance imaging analysis, econometrics, home price analysis, and many other domains. The phenomenon of sparsity is often found in nature, when a limited number of factors contribute significantly to the overall variation. Spatial regression models frequently use sparsity to indicate less complex computational and more superficial covariance structures. The spatial error model is a significant spatial regression model that focuses on the geographical dependence present in the error terms rather than the response variable. This study proposes an effective approach for estimating the vector of regression coefficients in the spatial error model, taking into consideration of the prior knowledge that some coefficients are insignificant and there is multicollinearity among the regressors. It also introduces pretest and shrinkage ridge estimators for spatial error regression models, evaluating their performance compared to traditional maximum likelihood estimators. It also assesses their efficacy using real-world data and bootstrapping techniques for comparison purposes.

This article deals with a finite-source stock-dependent stochastic inventory system with heterogeneous servers and retrial facility. The system can store up to S number of items, and the lifetime of each item follows an exponential distribution. The primary customers arrive in a finite waiting hall from a finite source and get service from the multi-channel (c-heterogeneous servers). The customer’s arrival rate depends on the present stock level. Those customers who arrive while the waiting hall is full enter into the finite orbit. Also, customers in the waiting hall can become impatient and enter the orbit. The (s,Q) ordering policy is adopted for the replenishment of stocks. In the steady state, the joint probability distribution of a number of items in the inventory, busy servers, customers in the waiting hall, and orbit is computed. The comparative numerical analysis of heterogeneous and homogeneous servers on the expected total cost, the average number of customers in the waiting hall and orbit, the fraction of successful retrial rate, and the average impatient customer rate are also carried out.

The aim of this paper is to study a semi-functional partial linear regression model (SFPLR) for spatial data with responses missing at random. The estimators are constructed by the kernel method, and some asymptotic properties such as probability convergence rates of the nonparametric component and asymptotic distribution of the parametric and nonparametric components are established under certain conditions. Next, the performances and the superiority of these estimators are presented and examined using a study on simulated data and on real data by carrying out a comparison between our semi-functional partially linear model with MAR estimator (SFPLRM), the semi-functional partially linear model with the full-case estimator (SFPLRC) and the nonparametric functional model estimator with MAR (FNPM). The results show that the proposed estimators outperform existing estimators as the number of random missing data increases.

This paper aims to contribute to refining the e-values for testing precise hypotheses, especially when dealing with nuisance parameters, leveraging the effectiveness of asymptotic expansions of the posterior. The proposed approach offers the advantage of bypassing the need for elicitation of priors and reference functions for the nuisance parameters and the multidimensional integration step. For this purpose, starting from a Laplace approximation, a posterior distribution for the parameter of interest only is considered and then a suitable objective matching prior is introduced, ensuring that the posterior mode aligns with an equivariant frequentist estimator. Consequently, both Highest Probability Density credible sets and the e-value remain invariant. Some targeted and challenging examples are discussed.

Empirical equations representing, interpolating and smoothing groups of measurement results by regression methods are in widespread use in metrology and other fields of science and engineering. Standard equations for the propagation of measurement uncertainties to values computed from such equations are available but suffer from a lack of general acceptance and are only infrequently applied in practice. One reason for the slow uptake is the lack of clear methods to account for systematic error. In this paper, uncertainty propagation equations in terms of covariance matrices are generalized to allow for systematic errors in least-squares regression, and effects of the resulting uncertainties are investigated analytically. A stochastic ensemble model of systematic error is proposed for the computation of the non-diagonal elements of the weight matrix of the Generalized Least-Squares method (GLS) from the measurement uncertainty. A GLS projector formalism is described which separates the effects of measurement scatter on values calculated by the equation from those on the related “residuals”, i.e., the residual errors in the fitted data. The same projectors act similarly on the associated uncertainties and covariance matrices and permit the effects of systematic errors on the simulation covariance to be quantified. It is demonstrated that, in order to include systematic errors in the uncertainty estimates of GLS, covariance matrices may be substituted by novel, specifically defined dispersion matrices that are not specified yet by the GUM[1]. Systematic measurement errors may be estimated from various sources; a particularly easy way suggested here involves analyzing the structure in the regression residuals. It is demonstrated that GLS projectors exhibit inherent features of “error blindness” also with respect to systematic errors. [1] GUM: Guide to the Expression of Uncertainty in Measurement, http://www.bipm.org/en/publications/guides

We present a new penalized method for estimation in sparse logistic regression models with group structure. Group sparsity suggests us to consider the Group Lasso penalty. Being different from penalized log-likelihood estimation, our method can be viewed as a penalized weighted score function method. Under some mild conditions, we provide non-asymptotic oracle inequalities promoting group sparsity of predictors. A modified block coordinate descent algorithm based on a weighted score function is also employed. The net advantage of our algorithm over the existing Group Lasso-type procedures is that the tuning parameter can be pre-specified. The simulations show that this algorithm is considerably faster and more stable than competing methods. Finally, we illustrate our methodology with two real data sets.

Financial time series and other human-driven, non-natural processes are known to exhibit fat-tailed outcome distributions. That is, such processes demonstrate a greater tendency for extreme outcomes than the normal distribution or other natural distributional processes would predict. We examine the mathematical expectation, or simply "expectation," traditionally the probability-weighted outcome, regarded since the seventeenth century as the mathematical definition of "expectation." However, when considering the "expectation" of an individual confronted with a finite sequence of outcomes, particularly existential outcomes (e.g., a trader with a limited time to perform or lose his position in a trading operation), we find this individual "expects" the median terminal outcome over those finite trials, with the classical seventeenth-century definition being the asymptotic limit as trials increase to this. Since such finite-sequence "expectations" often differ in values from the classic one, so do the optimal allocations (e.g., growth-optimal). We examine these for fat-tailed distributions. The focus is on implementation, and the techniques described can be applied to all distributional forms.

The distribution laws of various natural and anthropogenic processes in the world around us are stochastic in nature. The development of mathematics and, in particular, of stochastic modelling allows us to study regularities in such processes. In practice, stochastic modelling finds a huge number of applications in various fields, including finance and economics. In this work, some particular applications of stochastic processes in finance are examined in the conditions of financial crisis. More specifically, Autoregressive Integrated Moving Average (ARIMA) models and Modified Ordinary Differential Equations (ODE) models, which have been previously developed by the authors to predict assets’ prices of four Bulgarian companies, are validated on a time period during the crisis. Estimated rates of return are calculated from the models for one period ahead. The errors are estimated and the models are compared. The predicted return values with each of the two approaches are used to derive optimal risk portfolios based on the Markowitz Model. The resulting portfolios are compared in terms of distribution (weights of the stocks), risk and rate of return.

Measurements of systems taken along a continuous functional dimension, such as time or space, are ubiquitous in many fields, from the physical and biological sciences to economics and engineering. Such measurements can be viewed as realisations of an underlying smooth process sampled over the continuum. However, traditional methods for independence testing and causal learning are not directly applicable to such data, as they do not take into account the dependence along the functional dimension. By using specifically designed kernels, we introduce statistical tests for bivariate, joint, and conditional independence for functional variables. Our method not only extends the applicability to functional data of the Hilbert-Schmidt independence criterion (HSIC) and its d-variate version (d-HSIC), but also allows us to introduce a test for conditional independence by defining a novel statistic for the conditional permutation test (CPT) based on the Hilbert-Schmidt conditional independence criterion (HSCIC), with optimised regularisation strength estimated through an evaluation rejection rate. Our empirical results of the size and power of these tests on synthetic functional data show good performance, and we then exemplify their application to several constraint- and regression-based causal structure learning problems, including both synthetic examples and real socio-economic data.

This paper presents a new discrete counterpart of the mixture exponential distribution by utilizing the survival discretization method. The moment-generating function and associated moment measures are discussed. The distribution’s hazard rate function can assume increasing or decreasing forms, making it adaptable for diverse fields requiring count data modelling. The paper explores four distinct parameter estimation methods and assesses their performance through Monte Carlo simulations. The applicability of this distribution extends to time series analysis, particularly within the framework of the first-order integer-valued autoregressive process. Additionally, the paper explores quality control applications, addressing serial dependence challenges in count data encountered in production and market management. The performance of two distinct control charts, the cumulative sum chart and the exponentially weighted moving average chart, is evaluated for their effectiveness in detecting shifts in the process means under various models. A bivariate Markov chain approach is used to estimate the average run lengths of these charts, offering valuable insights for implementation. Design recommendations for achieving robustness in-control chart applications are provided. The effectiveness of the proposed models and charts is illustrated using a real data, demonstrating their practical superiority.

Random Forest (RF) is a widely used data prediction and variable selection technique. However, the variable selection aspect of RF can become unreliable when there are more irrelevant variables than relevant ones. In response, we introduced the Bayesian Random Forest (BRF) method specifically designed for high-dimensional datasets with a sparse covariate structure. Our research demonstrates that BRF possesses the oracle property, which means it achieves strong selection consistency without compromising efficiency or bias.

This article deals with the queueing-inventory system, composed of c junior servers, a senior server, two finite waiting halls, and an infinite orbit. On occasion, junior servers encounter challenges during customer service. In these instances, they approach the senior server for guidance in resolving the issue. Suppose the senior server is engaged with another junior server. The approaching junior servers await their turn in a finite waiting area with a capacity of c for consultation. Concerning this, we study the performance of junior servers approaching the senior server in the retrial queueing-inventory model with the two finite waiting halls dedicated to the primary customers and the junior servers for consultation. We formulate a level-dependent QBD process and solve its steady-state probability vector using Neuts and Rao’s truncation method. The stability condition of the system is derived, and the R matrix is computed. Optimum total cost has been obtained, and sensitivity analyses, which include expected total cost, waiting time of customers in waiting hall and orbit, number of busy servers, and fraction of successful retrial rate of the model, are done numerically.

In this article we introduce an extension of the Akash distribution. We use the slash 1 methodology to make the kurtosis of the Akash distribution more flexible. We study the general 2 density of this new distribution, some properties, moments, coefficients of asymmetry and kurtosis. 3 Statistical inference is performed using the methods of moments and maximum likelihood via the 4 EM algorithm. A simulation study is carried out to observe the behavior of the maximum likelihood 5 estimator. An application to a real data set with high kurtosis is considered, where it is shown that 6 the new distribution fits better than other extensions of the Akash distribution.

In this work, we propose a generalized partially functional linear model, which not only models the relationship between multiple scalar and functional predictors and response, but also automatically estimates the link function. Specifically, we use the functional principal component analysis method to reduce the dimensionality of functional predictors, estimate the regression coefficients using the maximum likelihood estimation method, estimate the link function using the method of local linear regression, iteratively obtain the final estimator, and prove the asymptotic normality of the estimator. The asymptotic normality is illustrated through simulation experiments. Finally, the proposed model is applied to study the influence of environmental, economic, and medical levels on life expectancy in China. In the study functional predictors are the daily air quality index, temperature, and humidity of 58 cities in 2020 and scalar predictors are GDP and number of beds in hospitals.

In this study, we present different methods of estimating fuzzy reliability of a two-parameter Rayleigh distribution via the maximum likelihood estimator, median first-order statistics estimator, quartile estimator, L-moment estimator, and mixed Thompson-type estimator. The mean-square error MSE as a measurement for comparing the considered methods using simulation through different values for the parameters and unalike sample sizes is used. The results of simulation show that the fuzziness values are better than the real values for all sample sizes, as well as the fuzzy reliability at the estimation of the Maximum

We introduce the lower chaos grade of a real-valued function F defined on the Markov triple (E,μ,Γ), where μ is a probability measure and Γ is the carré du champ operator. As an application of this concept, we obtain the better estimate of the four moments theorem for Markov diffusion generators worked by Bourguin et al (2019). For our purpose, we need to find the largest number except zero in the set of eigenvalues corresponding to its eigenfunction in the case where the square of a random variable F, coming from a Markov triple structure, can be expressed as a sum of eigenfunctions, We give some examples of eigenfunctions of the diffusion generators such as Ornstein-Uhlenbeck, Jacobi and Romanovski-Routh. In particular, two bounds, called the four moments theorem and fourth moment theorem respectively, will be provided for the normal approximation of the case where a random variable F comes from eigenfunctions of a Jacobi generator.

Positive recurrence for a single-server queueing system is established under generalised intensity conditions of service which does not assume existence of the density distribution function of service but a certain integral type lower bound as a sufficient condition. Positive recurrence implies existence of the invariant distribution and a guaranteed slow convergence to it in the total variation metric.

In this paper, we identify several relative ordering properties of the modified proportional hazard rate and modified proportional reversed hazard rate models. For this purpose, we use two well-known relative orderings, namely the relative hazard rate ordering and the relative reversed hazard rate ordering. The investigation is to see how a relative ordering between two possible base distributions for the response distributions in these models is preserved when the parameters of the underlying models are changed. We will give some examples to illustrate the results and the conditions under which they are obtained.

Financial technologies, stemming from the application of artificial intelligence to big data in finance, are continuously expanding, across different markets and financial services. While financial technologies bring many opportunities, such as reduced costs and extended inclusion, they also bring risks, among which cyber risks, which are constantly increasing and are difficult to measure. Among the difficulties in measurement lies the existence of interdependence among different cyber risks. The study of interdependence and possible contagion channels between cyber attacks to different institutions and economic sectors is indeed increasingly important to ensure economic and financial sustainability. Against this backdrop, this paper proposes a multivariate model for count time series of cyber risk events, in which the time-varying intensity parameter determining the probability that a cyber attack occurs evolves according to general autoregressive score models, taking both time and sectorial dependence into account. The model is particularly suitable for studying how the behaviors of different markets or sectors are interconnected and it constitutes a new approach to the multivariate analysis of count time series of cyber loss events.

In this paper, we investigate the cumulative historical entropy of the lifetime of an n-component coherent system under the assumption that all system components have failed at time t. Using the system signature, we can calculate the cumulative historical entropy of the system’s lifespan, which is an important aspect in determining a system’s predictability. This research produces a number of significant results, including expressions, bounds, and ordering aspects for this measure. In addition, we offer a method for finding a preferred system based on cumulative Kullback-Leibler discriminating information that is closely related to the parallel system. These findings provide important insights into the predictability and dependability of coherent systems, emphasizing the importance of this field.

This paper studies single change-point detection in the volatility of a class of parametric conditional heteroscedastic autoregressive nonlinear (CHARN) models. The conditional least-squares (CLS) estimators of the parameters are defined and are proved to be consistent. A Kolmogorov-Smirnov type-test for change-point detection is constructed and its null distribution is provided. An estimator of the change-point location is defined. Its consistency and its limiting distribution are studied in detail. A simulation experiment is carried out to assess the performance of the results which are also applied to two sets of real data.

The impacts of COVID-19 (Novel Coronavirus) epidemic cannot have been more severe, with the globe experiencing both economic and health crises. This study examines the trends and correlations in number of COVID-19-related deaths and number of COVID-19-infected patients in all 37 regions of Tamil Nadu state, India, in the month of August 2020 on the basis of panel regression model,

Most existing data synthesis methods are designed to tackle problems such as dataset imbalance, data anonymization and insufficient sample size. There is a lack of effective synthesis methods for the limited number of datasets which contain a large of features and unknown noise to expand the size of the dataset. We propose a data synthesis method, named Adaptive Subspace Interpolation for Sample Optimization (ASISO). The idea is to divide the original feature space into several subspaces with an equal number of samples, and then perform interpolation for the samples in the adjacent subspaces. This method can adaptively adjust the size of the dataset containing unknown noise, and the expanded data typically contain minimal error with actual. Moreover, it adjusts the structure of the samples, which can significantly reduce the proportion of samples with large errors. In addition, the hyperparameters of this method have an intuitive explanation and usually require little calibration. Experimental results on artificial data and benchmark data sets demonstrate that ASISO is a robust and stable method to optimize samples.

In this paper we study a special kind of finite mixture model. The sample drawn from the model consisits of three parts. The first two parts are drawn from specified distributions f1 and f2, while the third one is drawn from the mixture. A problem of interest is whether the two distributions f1 and f2 are the same. To test this hypothesis, we first define the regular location and scale family of distributions and assume that f1 and f2 are regular denisty functions. Then the hypothesis transforms to the equalities of the location and scale parameters, respectively. To utilize the information in the sample, we use Bayes’ theorem to obtain the posterior distribution and give the sampling method. Then we propose the posterior p-value to test the hypothesis. The simulation studies show that our posterior p-value largely improves the power in both normal and logistic cases while nicely controls the Type-I error. A real halibut dataset is used to illustrate the validity of our method.

We study the robustness of the cμ-rule for the optimal allocation of a resource consisting of one unreliable server to parallel queues with two different classes of customers. The customers in queues can be served with respect to a FCFS retrial discipline, when the customers at the head of queues repeatedly tries to occupy the server in a random time. It is proved that for the scheduling problem in the system without arrivals the cμ-rule minimizes the total average cost. For the system with arrivals it is difficult directly to prove the optimality of the same policy with explicit relations. We derived for an infinite-buffer model a static control policy that also prescribes for the certain values of system parameters the service exclusively for the class-i customers if both of queues are not empty with the aim to minimize the average cost per unit of time. It is also shown that in a finite-buffer case the cμ-rule fails.

An empirical investigation was carried out to study the relationship between the soil characteristics, including the Nitrogen (N), Phosphorus (P), Potassium (K), potential of Hydrogen (pH), Electrical conductivity (EC), Sulfur (S,), Zinc (Zn), Iron (Fe), Copper (Cu), Manganese (Mn) and Boron (B) using principal component analysis (PCA), Factor Analysis (FA), and Canonical Correlation analysis (CCA) for data reduction multivariate techniques. The first five main components accounted for 24.22%, 18.52%, 16.1%, 9.97%, and 9.27% of sample variances, respectively. All five principal components (PCs) accounted for 78.07% of the sample variance. The soil characteristics viz., P and K have highly dominated the first PC; the soil characteristics N and pH have dominated the second PC; K, EC, and Cu have dominated the third PC; Fe, Mn, and pH have dominated in the fourth PC; whereas the parameter Cu dominated the fifth PC. The first factor showed a strong negative loading on S and a strong favorable loading on Cu; P, K, and EC have substantial positive loadings in the second factor. Significant positive loadings on Mn, Fe, Cu, pH, N, and K are present in the third component. The fourth factor had a significantly positive pH, Fe, N, P, K, and B loadings. Fe, P, and B have much weight in the fifth factor. The soil characteristics viz., N, Zn, pH, K, Fe, and Mn have dominated the first PC; the soil characteristics B, P, S, Zn, and Fe have dominated the second PC; B and P have dominated the third PC whereas the only character Cu dominated the fourth PC. Cu and OC have much weight in the fourth factor. The canonical redundancies for dependent and independent sets are 12% and 17%, respectively. The Stewart-Love canonical redundancy index is 70.58%, which means that the first linear combination of the X-set explains 70.58% of the total variance in the Y-set.

To estimate the degree of quantum entanglement, it is important to understand the statistical behavior of functions of spectrum of density matrices such as von Neumann entropy, quantum purity, and entanglement capacity. These entangle- ment metrics over different generic state ensembles have been studied intensively in the literature. As an alternative metric, in this work we study sum of square root spectrum of density matrices, which is relevant to negativity and fidelity in quantum information processing. In particular, we derive the exact mean and vari- ance of sum of square root spectrum over the Bures-Hall generic state ensemble extending known results obtained recently over the Hilbert-Schmidt ensemble.

Distributions with bounded support show considerable sparsity over those with unbounded support, despite the fact that there are a number of real-world contexts where observations take values from a bounded range (proportions, percentages, and fractions are typical examples). For proportion modelling, a flexible family of two-parameter distribution functions associated with the exponential distribution is proposed here. Mathematical and statistical properties of the novel distribution are examined, including quantiles, mode, moments, hazard rate function, and its characterization. The parameter estimation procedure using the maximum likelihood method was carried out, and applications to environmental and engineering data were also considered. To this end, various statistical tests are used, along with some other information criterion indicators to determine how well the model fits the data. The proposed model is found to be the most efficient plan in most cases for the datasets considered.

Generalized Bayes is a Bayesian approach based on the learning rate parameter η. In this study, we examine the effect of parameter η on the estimation results considering joint type-I and type-II hybrid censored samples from k exponential populations. In addition to the learning rate parameter, we consider two loss functions, the Linex and general entropy loss functions in the Bayesian approach. Monte Carlo simulations are performed to compare the performances of the estimation results under losses and different values of η. An illustrative example is performed to study the effect of the learning rate parameter and the different losses with different parameters.

In this paper a more flexible extension of the Fréchet distribution is introduced. The new distribution is defined by means of the stochastic representation as the quotient of two independent random variables, a Fréchet distribution and the power of a random variable with uniform distribution in the interval (0,1). We will call this new extension the Slash Fréchet distribution and one of its main characteristics is that its tails are heavier than the Fréchet distribution. The general density of this distribution and some basic properties are determined. Its moments, skewness coefficients and kurtosis are calculated. In addition, the estimation of the model parameters is obtained by the method of moments and maximum likelihood. Finally, two applications with real data are performed by fitting the new model and comparing it with the Fréchet distribution.

In many research fields, statistical probability models are often used to analyze real-world data. However, data from many fields, such as the environment, economics, and health care, do not necessarily fit traditional models. New empirical models need to be developed to improve their fit. In this paper, we explore a further extension of the quasi-Lindley model. Maximum likelihood, least square error, Anderson-Darling algorithm, and expectation-maximization algorithm are four techniques for estimating the parameters under study. All techniques provide accurate and reliable estimates of the parameters. However, the mean square error of the expectation maximization approach was lower. The usefulness of the proposed model was demonstrated by analyzing a dynamical systems data set, and the analysis shows that it outperforms the other models in all statistical models considered.

In estimating logistic regression models, convergence of the maximization algorithm is critical, however, this may fail. Numerous bias correction methods for maximum likelihood estimates of the parameters have been conducted for cases of complete data sets and also longitudinal models. For binary response fixed effects panel data model, the conditional logit estimator is consistent for balanced data. When faced with missing covariates problem, researchers opt for various imputation techniques to make the data complete and without loss of generality consistent estimates still suffice asymptotically. For maximum likelihood estimates of the parameters for logistic regression in cases of imputed covariates, the optimal choice of an imputation technique which yields the best estimates with minimum variance is still elusive. The main aim of this paper is to examine the behaviour of the Hessian matrix with optimal values of the imputed covariates vector which will make the Newton-Raphson algorithm to converge faster through a reduced absolute value of the product of the score function and the inverse fisher information component. We focus on a method used to modify the conditional likelihood function through partitioning of the covariate matrix. We also confirm from the moduli of the Hessian matrices that the log likelihood of a panel data logistic model has a global maximum as the parameter estimates. Simulation results reveal that model based simulation perform better than classical imputation techniques yielding estimates with smaller bias and higher precision for the conditional maximum likelihood estimation of nonlinear panel models with single fixed effects.

Analyzing big data poses a great challenge for numerous researchers to explore the data structure. Dimension reduction methods can be used to reduce data dimensionality, taking it from occupying a high-dimensional space to existing in a lower-dimensional space while retaining as much information as possible. Principal Component Analysis is one of the most popular used to reduce the dimensional space. The bootstrap sample is obtained by randomly sampling n times with replacement from the original sample, the method provides easy tool to understand the interactive component and develop the process. There are not enough researches discussed the stability of PCA method using bootstrap method. In this paper, the bootstrap method is used to analyze the stability of PCA results and to estimate the number of PCA in efficient way. The method is used to estimate the number of PCA that needed to classify the data set, and the effectiveness of the discussed techniques is demonstrated through real data sets.

This paper focuses on a joint model to analyse longitudinal proportion and survival data. After transforming the longitudinal proportional data by a logit function, we adopt a partially linear mixed-effect model for it, in which nonlinear function of time is fitted via using B-splines technique and a centered Dirichlet Process Mixture Model (CDPMM) is specified for a general distribution of random effects. The survival data is assumed to a Cox proportional hazard model, the sharing random effects joint model is developed for the two types of data. Combining the Gibbs sampler and the Metropolis-Hastings algorithm, we propose a Bayesian Lasso (BLasso) method to simultaneously estimate unknown parameters and select important covariates. Simulation studies are conducted to investigate the finite sample performance of the proposed methods. An example from the MA.5 research experiment is used to illustrate the proposed methodologies.

Nonparametric density estimation for nonnegative data is considered in a situation where a random sample is not directly available but the data are instead observed from the length-biased sampling. Due to the boundary bias problem of the location-scale kernel, the approach in this paper is an application of asymmetric kernel. Two nonparametric density estimators are proposed. The mean integrated squared error, strong consistency, and asymptotic normality of the estimators are investigated. Some simulations illustrate the finite sample performance of the estimators.

In econometrics and finance, volatility modelling has long been a specialised field for addressing a variety of issues pertaining to the risk and uncertainties of an asset. This study presents a robust framework, through a step-by-step design, that is relevant for effective Monte Carlo simulation (MCS) with empirical verifications to estimate volatility using the Generalized Autoregressive Score (GAS) model. The framework describes an organised approach to the MCS experiment that includes "background (optional), defining the aim, research questions, method of implementation, and summarised conclusion". The method of implementation is a workflow that consists of writing the code, setting the seed, setting the true parameter a priori, data generation process, and performance assessment through meta-statistics. Among the findings, it is experimentally demonstrated in the study that the GAS model with a lower unconditional shape parameter value can generate a dataset that adequately reflects the behaviour of financial time series data, relevant for volatility estimation. This dynamic framework is intended to help interested users on MCS experiments utilising the GAS model for reliable volatility calculations in finance and other areas.

This study aims to apply a Recommendation System with Content Based Filtering method with Jaccard Similarity and Cosine Similarity algorithms on the E-Learning Platform. Recommendation systems deal with how to provide personalized recommendations to users efficiently. The Content Based Filtering method with Jaccard Similarity and Cosine Similarity algorithms can be used to calculate the similarity value between E-Courses on the E-Learning Platform. Implementation for Recommendation System using Google Colaboratory with Python programming language. In the application of the Recommendation System dataset, Coursera Free Dataset consists of 975 instances. The recommendation results use the Jaccard Similarity algorithm with an average similarity value of 0.3 while the value of Cosine Similarity with the average similarity value is 0.6 where the similarity value of Cosine Similarity is higher. Based on the results of the Mean Absolute Error in the low recommendation system, the average MAE value for all iterations of Jaccard Similarity algorithm is 0.013 and for the Cosine Similarity algorithm the average MAE value for all iterations is 0.014. This shows that the Recommendation System with Jaccard Similarity and Cosine Similarity algorithms can be used on the E-Learning Platform to provide efficiency solutions for personalized recommendations.

Asymmetry in the upper and lower tails is an important feature in modeling bivariate distributions. This article focuses on the log ratio between the tail probabilities at upper and lower corners as a measure of tail asymmetry. Asymptotic behavior of this measure at extremely large and small thresholds is explored with particular emphasis on the skew-normal copula. Our numerical studies reveal that, when the correlation or skewness parameters are around at the boundary values, some asymptotic tail approximations of the skew-normal copulas proposed in the literature are not suitable to compute the measure of tail asymmetry with practically extremal thresholds.

In recent decades, there have been numerous endeavors to develop a novel category of survival distributions possessing enhanced flexibility through the extension of existing distributions. This article constructs and validates the statistical properties of a novel survival distribution in order to obtain an alternative distribution that is suitable for analyzing survival data by presenting the novel mixture of the Fréchet distribution along with statistical properties such as the probability density function (PDF), cumulative distribution function (CDF), rth ordinary moment, skewness, kurtosis, moment-generating function, mean, variance, mode, survival function, hazard function, and asymptotic behavior, as well as constructing the estimators of the unknown parameter by employing the expectation-maximization (EM) algorithm, and simulated annealing. Additionally, the performance of the proposed estimators was compared with bias, mean squared errors (MSE), and simulated variances, and given an illustrative example of the proposed distribution to the survival data set in order to show that the proposed distribution is appropriate for the right-skewed data. This will be extremely advantageous in survival analysis.

In this paper, the problem of predicting future failure times based on a jointly type-II censored sample from k exponential populations is considered. The Bayesian prediction intervals and point predictors were then obtained. Generalized Bayes is a Bayesian study based on a learning-rate parameter. This study investigated the effects of the learning rate parameters on the prediction results. The loss functions of squared error, Linex, and general entropy were used as point predictors. Monte Carlo simulations were performed to show the effectiveness of the learning rate parameter in improving the results of prediction intervals and point predictors.

High-dimensional measurement error data are becoming more prevalent across various fields. Research on measurement error regression models has gained increasing attention due to the risk of drawing inaccurate conclusions if measurement errors are ignored. When the dimension p is larger than the sample size n, it is challenging to develop statistical inference methods for high-dimensional measurement error regression models due to the existence of bias, nonconvexity of objective function, high computational cost and many other difficulties. Over the past few years, some works have overcome the aforementioned difficulties and proposed several statistical inference methods. This paper mainly reviews the current development on estimation, hypothesis testing and variable screening methods for high-dimensional measurement error regression models and shows the theoretical results of these methods with some directions worthy of exploring for future research.

The aim of this study is to obtain the Bayes estimators and the maximum likelihood estimators (MLEs) for the unknown parameters of the Rayleigh Weibull (RW) distribution based on progres-sive type-II censored samples. The approximate Bayes estimators are calculated using the idea of Lindley and Tierney-Kadane's approximation method under the squared-error loss function when the Bayes estimators are not handed in explicit forms. In this study, the approximate Bayes esti-mates are compared with the maximum likelihood estimates in the aspect of the estimated risks (ERs) using Monte Carlo simulation. In addition, the coverage probabilities of the parametric bootstrap estimates are calculated. Real lifetime data sets belonging to the cancer types as bladder cancer, head and neck cancer, and leukemia are used to illustrate the emprical results belonging to the approximate Bayes estimates, the maximum likelihood estimates, and the parametric bootstrap intervals.

This study rolls out a robust framework relevant for simulation studies through the Generalised Autoregressive Conditional Heteroscedasticity (GARCH) model using the rugarch package. The package is thoroughly investigated, and novel findings are identified for improved and effective simulations. The focus of the study is to provide necessary simulation steps for volatility estimation that involve "background (optional), defining the aim, research questions, method of implementation, and summarised conclusion". The method of implementation is a workflow that includes writing the code, setting the seed, setting the true parameters a priori, data generation process and performance assessment through meta-statistics. This novel, easy-to-understand steps are demonstrated on financial returns using illustrative Monte Carlo simulation with empirical verification. Among the findings, the study shows that regardless of the arrangement of the seed values, the efficiency and consistency of an estimator generally remain the same as the sample size increases. The study also derived a new and flexible true-parameter-recovery measure which can be used by researchers to determine the level of recovery of the true parameter by the MCS estimator. It is anticipated that the outcomes of this study will be broadly applicable in finance, with intuitive appeal in other areas, for volatility estimation.

Control charts have been widely used for monitoring process quality in manufacturing and play an important role in triggering a signal in time when detecting a change in process quality. Many control charts in literature assume that the in-control distribution of the univariate or multivariate process data is continuous and not categorical. This research develops two exponentially weighted moving average (EWMA) proportion control charts for monitoring a process with multinomial proportions when considering both large and small sample sizes. For a large sample size , the charting statistic depends on the well-known Pearson χ2 statistic, and the control limit of the EWMA proportion chart is determined by an asymptotical chi-square distribution. For a small sample size, we derive the exact mean and variance of the Pearson χ2 statistic. Hence, the exact EWMA proportion chart is determined. The proportion chart can also be applied to monitor the distribution-free continuous multivariate process as long as each categorical proportion associated with specification limits of each quality variable is known or estimated. Lastly, we investigate the detection performance of the proposed EWMA proportion chart by numerical analyses. Real data analysis demonstrates the beneficial application of the proposed EWMA proportion charts.

In actuarial practice, the modeling of total losses tied to a certain policy is a non-trivial task. Traditional parametric models to predict total losses have limitations due to complex distributional features such as extreme skewness, zero inflation, multi-modality, etc., and the lack of explicit solutions for log-normal convolution. In the recent literature, the application of the Dirichlet process mixture for insurance loss has been proposed to eliminate the risk of model misspecification biases; however, the effect of covariates as well as missing covariates in the modeling framework is rarely studied. In this article, we propose novel connections among covariate-dependent Dirichlet process mixture, log-normal convolution, and missing covariate imputation. Assuming an individual loss is log-normally distributed, we develop a log skew-normal Dirichlet process to approximate the log-normal sum. As a generative approach, our framework models the joint of outcome and covariates, which allows to impute missing covariates under the assumption of missingness at random. The performance is assessed by applying our model to several insurance datasets, and the empirical results demonstrate the benefit of our model compared to the existing actuarial models such as the Tweedie-based generalized linear model, generalized additive model, or multivariate adaptive regression spline.

For order-of-addition experiments, the response is affected by the addition order of the experimental materials. Consequently, the main interest focuses on creating a predictive model and an optimal design for optimizing the response. Van Nostrand (1995) proposed the pairwise-order (PWO) model for detecting PWO effects. Under the PWO model, the full PWO design is optimal under various criteria but is often unaffordable because of the large run size. In this paper, we consider the D-, A- and M.S.-optimal fractional PWO designs. We first present some results on information matrices. Then, a flexible and efficient algorithm is given for generating these optimal PWO designs. Numerical simulation shows that the generated design has an appealing efficiency in comparison with the full PWO design, though with only a small fraction of runs. Several comparisons with existing designs illustrate that the generated designs achieve better efficiencies, and the best PWO designs and some selected 100% efficient PWO designs generated by the new algorithm are reported.

Polynomial recurrence bounds for a class of stochastic differential equations with a gradient type drift and an additive Wiener process are studied without Lyapunov functions.

Let {ξ1, ξ2. . . .} be a sequence of independent possibly differently distributed random variables defined on a probability space (Ω, F, P) with distribution functions {F_ξ1., F_ξ2 , . . .}. Let η be a counting random variable independent of sequence {ξ1, ξ2. . . .}. In this paper, we find conditions under which distribution function of randomly stopped sum Sη = ξ1 + ξ2 + . . . + ξη belongs to the class of generalized subexponential distributions.

We give two examples of the appearance of heavy-tailed distributions in applications to social sciences. Among these distributions are the laws of Pareto, Lotka, and some new ones. The examples are illustrated by constructing suitable toy models.

The multivariate Gaussian random fields with matrix-based scaling laws are widely used for inference in statistics and many applied areas. In such contexts interests are often in symmetry and in the rates of change of spatial surfaces in any given direction. This article analyzes the almost sure sample function behavior for operator fractional Brownian motion, including multivariate fractional Brownian motion. We obtain the estimations of small ball probability and the strongly locally nondeterministic for operator fractional Brownian motion in any given direction. Applying these estimates we obtain Chung's laws of the iterated logarithm for spatial surfaces of operator fractional Brownian motion. Our results show that the precise rates of change of spatial surfaces are completely determined by the self-similarity exponent.

In this paper, under the symmetric entropy and the scale squared error loss functions, we consider the maximum likelihood (ML) estimation and Bayesian estimation of the Shannon entropy and Rényi entropy of the two-parameter inverse Weibull distribution. In the ML estimation, the dichotomy is used to solve likelihood equation. In addition, the approximation confidence interval is given by the Delta method. Because the form of estimation results is more complex in the Bayesian estimation, the Lindley approximation method is used to achieve the numerical calculation. Finally, Monte Carlo simulations and a real data set are used to illustrate the results derived. By comparing the mean square error between the estimated value and the real value, it can be found that the performance of ML estimation of Shannon entropy is better than that of Bayesian estimation, and there is no significant difference between the performance of ML estimation of Rényi entropy and that of Bayesian estimation.

In the first part of this article, we discuss and generalize the complete convergence introduced by Hsu and Robbins (1947) to the r-complete convergence introduced by Tartakovsky (1998). We also establish its relation to the r-quick convergence first introduced by Strassen (1967) and extensively studied by Lai (1976). Our work is motivated by various statistical problems, mostly in sequential analysis. As we show in the second part, generalizing and studying these convergence modes is important not only in probability theory but also to solve challenging statistical problems in hypothesis testing and changepoint detection for general stochastic non-i.i.d. models.

The article proposes a method for determining the p-value of correlations in compositional data, i.e., those data that arise as a result of dividing the original values by their sum. Data organized in this way are typical for many fields of knowledge, but there is still no consensus on methods for interpreting correlations in such data. In a space closed by normalizing quantity, correlation coefficients behave differently than under normal conditions: their probabilities of occurrence do not coincide with those inherent in the standard scale of estimates. In the tens of the new millennium, almost all newly emerging methods for estimating correlation in compositional data began to require mandatory log-transformation of the variable values. In the method proposed here there are no log-transformations. We return to the early stages of attempting to solve the problem and rely on negative shifts in correlations in the multinomial distribution. In modeling the data, we use a hybrid method that combines the hypergeometric distribution with the distribution of any other law. During our work on the calculation method, we found that the number of degrees of freedom in compositional data measures discretely only when all the normalizing sums are equal and that it decreases when the sums are not equal, becoming a continuously varying quantity. Estimation of the number of degrees of freedom and the strength of its influence on the magnitude of the shift in the distribution of correlation coefficients is the basis of the proposed method.

L\'evy walks represent important modeling tools for variety of real life processes. Their natural scaling limits are known to be described by the so-called material fractional derivatives. So far these scaling limits were derived for spatially homogeneous walks, where Fourier and Laplace transforms represent natural tools of analysis. Here we derive the corresponding limiting equations in the case of position depending times and velocities of walks, where Fourier transforms cannot be effectively applied. In fact, we derive three different limits (specified by the way the process is stopped at an attempt to cross the boundary), leading to three different multi-dimensional versions of Caputo-Dzherbashian derivatives, which correspond to different boundary conditions for the generators of the related Feller semigroups and processes. Some other extensions and generalisations are analysed.

In this paper, we will extend the expected value of the function w.r.t the uniform probability measure on sets measurable in the Caratheodory sense to be finite for a larger class of functions, since the set of all measurable functions with infinite or undefined expected values forms a prevalent subset of the set of all measurable functions, which means "almost all" measurable functions have infinite or undefined expected values. Before we define the specific problem in section 2, we will outline some preliminary definitions. We'll then define the specific problem (along with a partial solution in section 3) to visualize the complete solution. Along the way, we will ask a series of questions to clarify our understanding of the paper.

Quantile regression model is widely used in variable relationship research of general size data, due to strong robustness and more comprehensive description of the response variables' characteristics. With the increase of data size and data dimension, there have been some studies on high-dimensional quantile regression under the classical statistical framework, including higher-efficient frequency perspective, which is however at cost of randomness quantification, or lower-efficient Bayesian method based on MCMC sampling. To overcome these problems, we propose the high-dimensional quantile regression with Spike-and-Slab Lasso penalty based on variational Bayesian (VBSSLQR), which can not only improve the computational efficiency but also measure the randomness via variational distributions. The simulation studies and real data analysis illustrate that the proposed VBSSLQR method is superior to or equivalent to other quantile and non-quantile regression methods (including Bayesian and non-Bayesian methods), and its efficiency is higher than any other method.