Computer Science and Mathematics

Abstract: When several sources bear on two hypotheses, Royall’s three evidential outcomes (strong for one, strong for the other, weak) leave no room for a fourth state: some sources are strong evidence for one hypothesis and others for the other. We integrate four verdicts (supported, refuted, undetermined, contradicted) in one likelihood-based report and state which error probability of each verdict is bounded in finite samples, which vanishes only asymptotically, and when “contradicted” stops describing a real conflict between source parameters. Spurious contradiction has probability at most \( m/c \) for \( m \) sources and per-source threshold \( c \), under any dependence, for any correctly specified family. With a truth midway between two simple hypotheses the contradiction probability tends to \( 1-2^{1-m} \); a guarded rule keeps it at most \( m\Phi (-\sqrt{2\ln\,c}) \) for marginally Gaussian sources with any common mean and any dependence, and at most \( m(m-1)\Phi (-2\sqrt{\ln\,c}) \) under pairwise joint normality with non-negative covariances. For composite hypotheses an integrated-over-maximised likelihood ratio gives the bound \( m/c \) for heterogeneous parameters within one hypothesis. A preregistered simulation confirmed the bounds. In the Bacillus Calmette–Guérin (BCG) vaccine trials the verdict for “risk ratio at most 0.5” against “at least 1” is “contradicted” under normal, binomial-profile and conditional likelihoods; because these hypotheses leave an indifference zone, this does not establish that the trials’ true effects conflict.

Abstract: Ranked set sampling (RSS) helps when ranking units is cheap but measuring them is expensive, because it lets you measure only the units whose ranks tell you the most. This paper looks at a variant called except extreme ranked set sampling (EERSS), which drops the smallest and largest unit from every set and keeps only the middle ranks, and asks how well it estimates the cumulative distribution function F(x). We first derive the plain empirical CDF estimator under EERSS, along with its exact bias, mean squared error, and efficiency relative to simple random sampling (SRS). Unlike RSS, this estimator is not unbiased in general; it is exactly unbiased only at the point where F(x)=1/2, no matter what the population looks like. To fix this, we build a maximum likelihood estimator instead, treating each indicator of whether a ranked unit falls below x as a binomial outcome with a known probability tied to its rank. This estimator needs no assumption about the population's shape, and its Fisher information gives both an asymptotic variance and a confidence interval for F(x). A numerical comparison then puts both versions of EERSS side by side with SRS and RSS, in terms of bias, MSE, and relative efficiency, across several sample sizes and values of F(x). The clearest finding is this: the maximum likelihood EERSS estimator is the best choice near the centre of the distribution, clearly beating RSS and SRS there, while RSS remains the better choice in the tails. The plain empirical EERSS estimator, on the other hand, should generally be avoided, since its bias never shrinks no matter how much data is collected; switching to maximum likelihood is what actually makes EERSS worth using. A real dataset of combined cycle power plant measurements confirms that this pattern holds in practice, not just in theory.

Abstract: We study the martingale problem for two mutually competing superprocesses in independent Brownian random environments. The competition rates depend on the spatial position and both population measures. Under suitable assumptions, we establish well-posedness for every pair of finite initial measures and derive moment estimates for the total masses. Existence is proved by constructing branching particle systems and establishing tightness of their empirical measure processes. Uniqueness in law follows from pathwise uniqueness for the system of stochastic partial differential equations satisfied by the associated distribution-function-valued processes.

Abstract: We match independent uniform samples from the squares [0, 1]² and [2, 3] × [0, 1] by minimizing total Euclidean distance. The transport problem between the two uniform distributions has infinitely many optimal plans. We prove that, as the sample size grows, the empirical optimal plans converge weakly in probability to a single deterministic plan. This plan preserves height and has the same joint distribution of paired horizontal coordinates at every height. We prove convergence by establishing uniqueness of the stationary local matching obtained after rescaling and identifying the selected horizontal coupling from its source-Palm distribution.

Abstract: Fix \( \vartheta^2=1/m \), \( m\in\mathbb{N} \), and let \( M_n \) be the maximum of the first \( n \) digits of the stationary \( \vartheta \)-expansion process under its invariant measure. Sebe, Lascu and Selmi established the corresponding Fréchet-type extreme-value law together with quantitative error estimates. We prove a single explicit polynomial Kolmogorov bound that is uniform over the full level range \( y>0 \): \( \sup_{y>0}\left|\gamma_\vartheta(M_n\le ny)-\exp(-c_\vartheta/y)\right| \le C_\vartheta\frac{(\log n)^2}{\sqrt n}. \) Here \( c_\vartheta=1/\log(1+\vartheta^2) \). On every fixed range \( y\ge\delta>0 \) the rate improves to \( O_{\vartheta,\delta}((\log n/n)^{1/2}) \). As an intermediate step we isolate an elementary blocking estimate for exponentially \( \psi \)-mixing stationary sequences which remains uniform while the Poisson parameter grows as \( O(\log n) \). For the \( \vartheta \)-digit process, the exact marginal tail allows us to reach the boundary scale \( y\asymp1/\log n \); monotonicity then closes the remaining small-level range.

Abstract: Pilot samples arise naturally in two-stage statistical procedures, where preliminary observations are used to estimate an unknown optimal allocation before the remaining observations are assigned. The central question is the choice of pilot size. A small pilot gives poor estimation of the optimal allocation, while a large pilot commits too many observations before that allocation is known. Under allocation MSE and regular parametric estimation, these two effects lead to a sharp balance. The pilot size grows as the two-thirds power of the total sample size. The square-root pilot is first-order adequate, whereas the two-thirds rule gives the higher-order balance between estimation and early commitment. Four pilot-size regimes are considered to show the effect of choosing the pilot below, at, or above this order. The result is general and does not depend on a particular distribution. Exponential lifetime populations are used as an application for which the optimal allocation can be obtained explicitly. Reliability and medical examples illustrate implementation of the rule.

Abstract: Three-arm binary trials may require low-dose benefit over control and additional high-dose benefit. We evaluate posterior support, false-declaration control and power when both response-probability differences must reach positive minima and at least one exceeds its target. Conditioning on the shared low-dose probability under independent binomial sampling and independent beta priors gives the posterior probability of meeting this requirement. Probabilities of complete or partial attainment and expected shortfalls describe uncertainty. For tables retaining declarations when control responder counts decrease or high-dose counts increase, maximization of false-declaration probability over all configurations failing the requirement reduces to three one-dimensional paths. Certified bounds keep the saved tables' maximal false-declaration probability below 2.5%. With illustrative thresholds and uniform priors, the joint Bayesian rule's power was at most 1.3% across 24 acceptable profiles exceeding only one target per design (250 or 386 participants). Method-independent bounds constrain fixed-sample count-based rules with the same error control. At 30% control response, 200 low-dose participants, low-dose effects between a minimum and target of 7 and 10 percentage points, and an increment above 5 points, power cannot exceed approximately 14%, regardless of other group sizes. JADE MONO-2 counts illustrate retrospective reporting. The results distinguish satisfying a requirement from reliably establishing it.

Abstract: Correlation-based criteria are often used to select transformations that improve linear association on a transformed scale. This paper studies a structural limitation of such criteria in general parametric transformation families. We show that Pearson correlation is locally insensitive to the affine component of any infinitesimal perturbation of a transformed variable: the local derivative depends only on the residual component after projection onto the span of the constant function and the current transformed variable. We call this phenomenon affine invisibility. It provides a general mechanism for weak local discrimination and, when low visibility persists over parameter regions, for weak identification under correlation-based selection. The theory is developed for general smooth transformation paths and specialized to log-polynomial transformation classes. The Box–Cox transformation is reinterpreted as a one-dimensional path in function space whose tangent directions are combinations of log-polynomial components. Analytical illustrations quantify scale-free local visibility for log-polynomial directions and along the Box–Cox path, providing a concrete instance of the broader geometric framework. The results suggest that correlation maximization should be accompanied by local-visibility and near-optimality diagnostics before selected transformation parameters are interpreted as well identified.

Abstract: Comparing central tendency across groups in complex survey data is challenging when outcome variables are skewed or contain outliers. While Mood’s median test provides a robust nonparametric solution under simple random sampling, it is not directly applicable to data arising from complex survey designs. This article proposes a design-based extension that integrates weighted median estimation with Rao–Scott adjusted tests of independence.The method replaces sample counts with estimated population totals derived from survey weights and uses design-consistent variance estimation to account for clustering, stratification, and unequal probabilities of selection. A Rao–Scott correction is then applied to obtain valid inference. The approach is demonstrated using data from a WHO STEPS survey, highlighting its practical utility in epidemiological applications.

Abstract: Statistical analysis usually begins from recorded values treated as draws from a distribution. Those values acquired their numerical form through processes—choice of unit, rounding and coding, biological, arithmetic and institutional constraints, composition from other quantities, aggregation and ordering—that the abstraction X₁, . . . , X_n ∼ F leaves implicit. This article develops a construction-aware identification framework for recorded numerical data. An upstream state Z passes through a construction map indexed by a regime M to give the recorded value X, which an analyst transforms into a representation P(X) on which a diagnostic is computed. The framework makes the construction regime an explicit inferential target alongside the upstream state and shows that the two identification problems can move in opposite directions: a rounding rule can be ignorable for inference about a mean and yet identifiable from terminal digits. It places the analyst’s representation on the same chain as the recording process, so that a distinction the recorded values carry can be destroyed before any diagnostic is computed. Formal results characterize when construction regimes are observationally distinguishable from recorded values, when support alone can exclude a regime, and when a representation preserves or destroys that distinction. A locating vocabulary—unitization, quantization, constraint, composition, aggregation—says where numerical structure entered a record, and worked cases from temperature archives, survey blood pressure, the genetic code, check digits, marathon qualifying standards, seawater sound speed, tide gauges and federal spending records show the framework operating in each class. The practical consequence is a change in what a numerical diagnostic reports: not an anomaly flag, but whether the construction regime is identified, partially identified, excluded, or not identified by the information at hand.

Abstract: Generalizing a distribution can be achieved by inserting one or more new parameters to the baseline distribution. The author deliberates this methodology to build, Weibull Median Based Unit Rayleigh {Y} family of distributions. The author applies the approach of the T-X{Y} family to add more than one parameter to the baseline distribution Median Based Unit Rayleigh (MBUR) defined on the unit interval with one parameter b > 0. The generator distribution is the Weibull distribution with the two parameters, c & ξ > 0, the shape and scale parameters respectively as will be explained in the paper. The author consumed different link distributions or Y distributions in the form of the quantile function of this link distribution to link the baseline distribution, the MBUR, to the generator distribution, the Weibull distribution. The link distributions deployed are the Lomax, Dagum, exponential, exponentiated exponential and the Log Logistic distributions. This technique results in new different distributions with different number of parameters, as will be discussed in the paper. For each new distribution, the author derived the PDF, CDF, quantile and hazard functions as well as the raw moments. The author expands some of the PDFs in a linear series expansion that is simple and does not require special function. This makes the computational statistics for the raw moments easier and simpler. The supplementary material associated with the paper contains figures for the PDF, CDF and the Hazard functions. Real data analysis reveals correlation between the parameters of the new distributions. The author clarified this by comparing the fitness of the new distributions to the same dataset, rate of recovery from COVID 19 in Spain, to wonder whether this correlation is data dependent or distribution dependent or mix and interaction between both the data and the distribution.

Abstract: Air quality indices, severity ratings, and similar series come as ordered categories, yet standard forecasting methods produce continuous predictions. We propose a two-stage approach using NeuralProphet, which decomposes a series into trend, seasonality, and autoregressive components. In the first stage, NeuralProphet is fit to the ordinal labels treated as continuous, yielding a decomposition and a continuous forecast. In the second, ordered thresholds mapping it to categories are optimized on a validation segment, replacing naive rounding. We evaluate on a controlled simulation and two real series, Beijing air quality and electricity load in a Ball State University building. On the simulation, the method recovers the latent ordering and generating components despite being fit on discretized labels, and threshold optimization improves the quadratic weighted kappa over rounding by 0.05 to 0.08, and by more when a miscalibrated trend leaves the ordering intact but the level off. On the real series it leads at every horizon, reaching a kappa of 0.57 on air quality and 0.89 on building load at 72 hours, and the gain grows with the horizon on air quality. The fitted components match each series’ structure. The approach is therefore an effective, interpretable framework for ordinal forecasting without specialized machinery.

Abstract: This study extends fractal boundary analysis from national borders to continental and subcontinental landmasses through a 29-unit hierarchical framework comprising six continents and 23 subcontinents, defined under a modified UN M49 structure with study-specific connectivity rules. Each geographical unit was constructed and measured directly from its external perimeter, so parent-continent estimates represent distinct boundary geometries rather than averages or weighted combinations of constituent subcontinental dimensions. Adaptive box-counting (BC) and divider (DIV) estimators were applied under a common finite-scale procedure, followed by descriptive scale-stratified analysis of boundary complexity, parent–subcontinent contrasts, cross-method correspondence, method sensitivity, and equal-weight consensus fractal dimensions (FDs). Continental mean FDs exceeded subcontinental means under both methods (BC: 1.160842 vs. 1.136353; DIV: 1.172651 vs. 1.141075). Cross-method agreement was strong at both scales (continents: Pearson r = 0.9876, Lin CCC = 0.9624; subcontinents: Pearson r = 0.9822, Lin CCC = 0.9743). Greenland and Northern Europe occupied the upper end of the hierarchy, whereas Southern Africa formed the lower extreme. Despite this overall agreement, substantial within-family heterogeneity and localized method-sensitive differences remained. The resulting Continental Atlas provides a reproducible hierarchical framework for characterizing macro- scale boundary complexity while preserving geographical scale, direct perimeter construction, and estimator-specific information.

Abstract: This paper examines the performance of the Homogeneously Weighted Moving Average on Wilcoxon Rank- Sum statistic (W-HWMA) and the Enhanced W-HWMA (W-EHWMA) control charts under different distributional conditions. The enhanced scheme adds an extra smoothing-parameter weighting to the variance structure to increase sensitivity to small and moderate size changes in processes without affecting the long-memory characteristics of weighted-average charts. The performance of the proposed chart was investigated in terms of the run length properties using comprehensive Monte Carlo simulations under the Normal, Student-t, Exponential and Weibull distributions. The findings indicate that the distributional robustness of both charts is high, but the novel W-EHWMA chart is more effective than the W-HWMA chart and across the investigated distributions and smoothing parameters, the proposed scheme generally produced lower ARL₁ and EARL values than the W-HWMA scheme, with the magnitude of improvement depending on the distribution, shift size and smoothing parameter. The superior design also produces narrower and more stable control limits that favours earlier out-of-control detection capability for small deviations in the process irrespective of the type of distribution. The simplicity and effectiveness in the practicality of the proposed W-EHWMA chart offer a significant advantage over the existing nonparametric charts.

Abstract: This study establishes a harmonized worldwide statistical atlas of principal-landmass national border complexity, addressing the absence of any dataset estimating their fractal dimensions under one common dual-method framework. Box-counting and divider fractal dimensions were estimated for 197 countries and territories from consistently processed GADM 4.1 geometries and evaluated at continental and global levels. Most principal-landmass borders proved measurably fractal yet occupied a comparatively low complexity band, departing from the linear benchmark while remaining far from plane-filling geometry: the worldwide mean dimensions were 1.0959 and 1.0975, respectively, with no evidence of a systematic paired difference (mean difference = −0.0015). Cross-method correspondence was strong (Pearson r = 0.9155; Lin CCC = 0.9069), although Bland–Altman and rank analyses identified modest magnitude-dependent disagreement and a limited set of method-sensitive countries. The equal-weight consensus fractal dimension averaged 1.0967 worldwide, ranging from 1.0170 for Somalia to 1.3615 for Norway, and continental consensus averages were highest in North America (1.1259) and Europe (1.1150) and lowest in Africa (1.0652). With all source geometries, candidate scales, and estimates archived for replication, the country-level values form a reproducible baseline for comparative, sensitivity, and time-indexed analyses of national-border complexity.

Abstract: Extreme Gaussian tail probabilities arise in a wide range of statistical applications, including large--scale hypothesis testing, meta--analysis, and risk assessment. Although the normal distribution is analytically well understood, reliable numerical evaluation of the tail probability \[ 1-\Phi(Z) \] becomes challenging for large values of Z, due to the combined effects of numerical underflow, truncation of the infinite integration domain, and discretization error in numerical integration. In this work, we present a theoretically grounded framework for evaluating extreme Gaussian tail probabilities under standard double–precision arithmetic. Rather than introducing a new closed–form approximation, we reformulate the problem explicitly as a controlled numerical integration task and decompose the total numerical error into three independent components: underflow, truncation error, and numerical integration error. Central to this formulation are explicit analytical lower and upper bounds, F(Z) and G(Z), which satisfy \[ F(Z) \le 1-\Phi(Z) \le G(Z) \] and provide a rigorous enclosure of the true tail probability. Within this framework, numerical underflow is avoided by scale separation and logarithmic representations, truncation error is controlled by selecting a finite integration range based on the analytical bounds, and numerical integration error is regulated through standard quadrature error estimates. Numerical experiments show that, as \(Z\) increases, the effective integration range shrinks and the required subdivision number remains moderate, explaining the empirical robustness of Gaussian tail evaluations observed in practice. The proposed formulation does not aim to compete with existing highly optimized implementations based on minimax rational approximations. Instead, it provides a transparent theoretical interpretation of why such implementations remain stable and accurate even in extreme regimes. By explicitly connecting classical analytical bounds with numerical error control, this work offers a principled perspective on Gaussian tail evaluation and clarifies the structure underlying long--standing numerical practices.

Abstract: This paper analyzes a GI/M/1/N retrial queue with multiple working vacations, Bernoulli feedback, and impatient customers with retention. Customers who find the server busy do not wait in a physical queue; they join a retrial orbit in which the total retrial attempt rate is proportional to the orbit size (the classical linear retrial policy, rη). Using the supplementary variable technique combined with an embedded Markov chain at arrival epochs, we derive the steady-state distributions at pre-arrival and arbitrary epochs, and we obtain closed-form expressions for the matrix exponentials required for deterministic, Erlang, and hyperexponential inter-arrival times. A degenerate-limit result shows that, as the retrial rate tends to infinity, the model reduces in distribution to the classical GI/M/1/N waiting-line queue of Guendouzi and Saadaoui; the numerical validation of Algorithm 1 reported in this paper is instead performed against an independent discrete-event simulation. We derive consistent performance measures (including a flow-balance identity usable as a numerical check), study the asymptotic behavior as the capacity N→∞, develop an economic model, and optimize it with particle swarm optimization (PSO) together with comparative GA and SA experiments and a robustness analysis of the cost parameters. Numerical illustrations highlight the impact of the system parameters on performance and profitability, with motivating applications to 6G networks, cloud computing, and smart healthcare.

Abstract: The cumulative distribution function (CDF) of the standard normal distribution plays a central role in statistical hypothesis testing, meta-analysis, and many applications involving the conversion of \(Z\)-scores into \(p\)-values. In this work, we introduce the normal-tanh coordinate \[ L(x)=\operatorname{atanh}\!\left(2\Phi(x)-1\right), \tag{1} \] where \(\Phi(x)\) denotes the standard normal cumulative distribution function. The normal-tanh coordinate transforms the bounded normal CDF into an unbounded odd function and provides a natural target for polynomial approximation. We derive cubic and quintic approximations of \(L(x)\) using two different approaches: a Taylor expansion and a structure-preserving approximation based on the differential equation satisfied by the Mills ratio. A striking result is that both approaches yield exactly the same cubic coefficient, \[ a_3=-\frac{1}{6}+\frac{2}{3\pi}, \tag{2} \] whereas they diverge at quintic order. Numerical experiments show that the ODE-preserving quintic approximation provides a better approximation over a wider range of \(x\), while the Taylor approximation remains optimal in the immediate neighborhood of the origin. Motivated by this representation, we introduce a logistic-type family of probability distributions defined by \[ F(x)=\frac{1}{2}+\frac{1}{2}\tanh\!\left(f(x)\right). \tag{3} \] Within this framework, both the logistic and the standard normal distributions arise naturally as special cases. The cubic approximation defines a valid member of this family and possesses closed-form expressions for its cumulative distribution function, probability density function, and quantile function. By contrast, the ODE-preserving quintic approximation improves approximation accuracy but does not satisfy the global conditions required for membership in the logistic-type family. The coefficient \(-\frac{1}{6}+\frac{2}{3\pi}\) is also remarkably close to the empirical cubic coefficient used in the GELU approximation, suggesting a possible connection between the local structure of the normal distribution and the empirical success of GELU-type transformations.

Abstract: We study a critical homogeneous continuous-time Markov branching-immigration system whose offspring and immigration generating functions are regularly varying at the point~1 of orders \(1+\nu\) and \(\delta\) with \(\nu,\delta\in(0,1]\), the slowly varying factors being controlled through their index functions. Two numbers govern the classification: the difference \(\gamma=\delta-\nu\) of the two orders and the limit \(\Lambda_{\infty}\) of the ratio of the immigration factor to the offspring one. We classify the state space completely: positive recurrent for \(\gamma>0\), transient for \(\gamma< 0\), and on the boundary \(\gamma=0\) null recurrent or transient according as \(\Lambda_{\infty}\le{\nu}\) or \(\Lambda_{\infty}>\nu\), which at \(\nu=\delta=1\) reduces to the classical threshold. In each regime we identify the limiting generating function of the normalised transition functions together with the exact rate of convergence to it, uniformly in the argument on compact subsets of [0,1), the transient rate under two further inequalities on the parameters; and we carry every rate to the individual transition probabilities, with no monotonicity hypothesis, by a complex-analytic argument resting on a factorisation that criticality forces. The constants are made accessible by an exact integral identity for the survival probability of the system without immigration, which also yields a three-term expansion of that probability in closed form. A numerical section on explicit models confirms the constants and the rates.

Abstract: In this paper we examine the relative influence of structure learning algorithms and scoring functions in Bayesian networks for healthcare applications. We compare Hill Climbing, Tabu Search, and Greedy Equivalence Search (GES) using the Bayesian Information Criterion (BIC), Bayesian Dirichlet equivalent uniform (BDeu), and K2 scoring functions on three healthcare datasets. On the full data, all three algorithms produced almost identical structures under BIC (Structural Hamming Distance [SHD] = 0 or 1). Bootstrap results showed a different pattern: Hill Climbing and Tabu Search remained identical in almost all resamples (median SHD = 0.0), whereas GES differed more from both (median SHD = 3.0–7.0). Changing the scoring function substantially altered the structure in two datasets but not the third. Overall, scoring functions produced larger differences than algorithms (median SHD = 8.5 vs. 3.0), although these values combine bootstrap and full-sample comparisons. Checks using different redundancy weights and discretization levels did not change the main pattern. Scoring functions were generally the larger source of disagreement, while algorithm agreement was less stable under resampling. Further datasets are needed to assess whether these findings generalize.