A Log-Dagum Weibull Distribution: Properties and Characterization AneeqaKhadima,

AneeqaKhadima, Aamir Saghira, TassadaqHussaina aDepartment of Mathematics, Mirpur University of Science and Technology (MUST), Mirpur10250 AJK, Pakistan aneeqa89@gmail.com; aamirstat@yahoo.com; tassaddaq.stat@must.edu.pk Abstract Developments of new probability models for data analysis are keen interest of importance for all fields. The log-Dagum distribution has a prominent role in the theory and practice of statistics. In this article, a new family of continuous distributions generated from a log Dagum random variable called the log-Dagum Weibull distribution is proposed. The key properties of the proposed distribution are derived. Its density function can be symmetrical, left-skewed, right-skewed and reversed-J shaped and can have increasing, decreasing, bathtub hazard rates shaped. The model parameters are estimated by the method of maximum likelihood and illustrate its importance by means of applications to real data sets.


Introduction
Statistical distributions are extensively used in literature for modelling and forecasting real life phenomena. The recent literature has suggested several ways of extending well-known distributions. There has been an increased interest in defining new classes of univariate continuous distributions by introducing one or more additional shape parameter(s) to the baseline distribution. This induction of parameter(s) has been proved useful in exploring tail properties and also for improving the goodness-of-fit of the generator family. The well-known families are: the beta-G [9], Kumaraswamy-G [6], McDonald-G [3], Gamma-X [2],Gamma-G (type 1) [18], Gamma-G (type 2) [15], Gamma-G (type 3) [17],Log-Gamma-G [4],Logistic-G [16],Exponentiated Generalized-G [7], Transformed-transformer [2], Exponentiated T-X [2], Weibull-G [5], etc.
Where W[G(x)] satisfies the condition (1). The pdf corresponding to (2) is given by In Table 1, we provide the W [G(x)] functions for some members of the T-X family of distributions. The aim of this paper is to propose a new family of continuous distributions, called the log Dagum Weibull distribution, and to study some of its mathematical properties.

The Log Dagum Weibull Distribution
A random variable T has the log Dagum distribution with shape parameter β > 0 and λ > 0 if it's cumulative distribution function (cdf) is given by and its corresponding probability density function (pdf) can be expressed as Let ( ) and ⃑ ( ) = 1 − ( ) be the baseline cdf and survival function (sf) by replacing ) and ( ) with (5) in equation (2), we define the cdf of the Log Dagum-X family by The Log Dagum family pdf is expressed as Henceforth, we denote by X a random variable having density function (7). The basic motivations for using the Log Dagum-x family in practice are to construct heavy tailed distributions that are not longer-tailed for modelling real data, to generate distributions with symmetric, left-skewed, right-skewed and reversed-J shaped, to define special models with all types of the hazard rate function (hrf), to provide consistently better fits than other generated models under the same baseline distribution. The fact is well-demonstrated by fitting the log Degum Weibull distribution to two real data sets. However, we expect that there are other contexts in which the LX special models can produce worse fits than other generated distributions. Clearly, the results indicate that the new family is a very competitive class to other widely known generators with one extra shape parameter.
The corresponding cumulative density function (cdf) probability density function (pdf), hazard function (hrf) and survival function are given as [ Figure 1 about here.] Figure 1 gives the plots of the cumulative distribution function of the LDW distribution.
The plots of this figure shows that for fixed and and changing the curve stretch out insignificantly towards right as increases. However, for fixed and and changing the curve stretch out towards right significantly as increases. And [ Figure 2 about here.] Plots of Figure 2 display the density functions of the LDW distribution. Figure 2 portrays that changing against the fixed and the density function decreases.
but changing against the fixed the nature of the curve towards right as increases, however in case of changing with fixed and shift the curve towards left.
[ Figure 4 about here.] The graph of survival function increases for different values of parameters then suddenly starts gradually decreases and converges to zero.
[ Figure 3 about here.] Hazard function is a significant indicator for observing the declining circumstance of a product which ranges from increasing, decreasing, bathtub (BT) shapes. So in this regard Figure 3 speaks out itself and justifies the potential of the model. Moreover, the hazard function plots in Figure 3 also portray the declining circumstance of the product as time increases in terms of impulsive spikes at the end of either increasing or decreasing hazard rate. This implies that the hazard function is sensitive against different combinations of the parameters as time changes, which seems to be a refine image of non stationary process and hence the hazard curve does not remain stable as times passes. Moreover, Figure 3 displays increasing, decreasing bathtub hazard shapes.

Shape of hazard function:
Shape of the density function can be described analytically .the critical point of the LDW density are the root of the equation There may be more than one root.

Some Statistical Properties
In this section, we study some statistical properties of the LDW distribution, including Rth moments, L-moments quantile function and order statistics.

Moments of LDWD.
Let X is a particularly continuous non-negative random variable with PDF ( ) , and then the R ordinary moment of the (LDW) distribution is given by:

Quantile function
The quantile function is another way of describing a probability distribution. It can also be called the inverse cdf. It can be used to generate random samples for probability distributions and thereby can serve as an alternative to the pdf. In general, it is given as:

Ordered Statistic:
The pdf of the jth order statistic for a random sample of size n from a distribution function ( ) and an associated pdf ( ) is given by: where ( ) and ( )are the pdf and cdf of the LDWD, respectively. The pdf of the jth order statistics for a random sample of size n from the LDW distribution is, however, given as So, the pdf of minimum order statistics is obtained by substituting = 1 we have: While the corresponding pdf of maximum order statistics is obtained by making the substitution of = in above equation

Entropies
Entropy is the measure of uncertainty. It is actually a concept of physics.

Characterization
In order to develop a stochastic function for a certain problem, it is necessary to know whether function fulfils the theory of specific underlying probability distribution, it is required to study characterizations of specific probability distribution. Different characterization techniques have developed. Glanzel (1987Glanzel ( , 1988 and 1990), Hamedani . This is the cdf of LDW distribution.

Characterization based on reversed hazard Function:
For random variable having LDW distribution with hazard rate function we obtain the following equation After manipulation, integrating and simplifying, we obtain as ( ) = 1 + − 1

Characterization through Ratio of Truncated Moments
In this section, we characterize WD distribution using Theorem 1 (Glanzel; 1987) on the basis of simple relationship between two functions of X. Theorem 1 is given in appendix A.

Proof:
For random variable X having LDW distribution with pdf (9) and cdf (8), we proceeds as Therefore in the light of Theorem 1, X has pdf (9)

Maximum Likelihood Estimation
Several approaches for parameter estimation have been proposed in the literature but the maximum likelihood method is the most commonly employed.
Let , , … , be a random sample of size n of the LDW distribution then the total log-likelihood (LL) function is given This section deals the simulation study. In proposed model we generated random variables by using CDF of LDWD with four different value of parameters for n=25,50,100,200.
Parameters are estimated with method of MLE by using each generated random variable.
In statistical study, bias states to the tendency of a measurement process to over or under The results are reported in Tables 1 and 2 . [

Evaluation Measures and Practical Data Examples
We illustrate the usefulness of the Log Dagum Weibull distribution and compare the results with the WD GD LD EED and NEED distributions by means of four real data sets. , one of which is data of leukaemia-free survival times of 50 patients with Autologous transplant obtained from [11] and the second data set contains Lifetime of 50 devices [13], Third data set consists of 100 uncensored data on breaking stress of carbon fibres (in Gba) [32], fourth data consist times to failure of eighteen electronic devices [33] In this section we illustrate the usefulness of the Log Dagum Weibull distribution .We estimate the unknown parameters of LDWD using MLE method and compare the log likelihood with some other distributions including EED, WD,GD,NEED and LD. We will check goodness of fit of our model with some test statistics like AD test, CVM test, K-S test and p-value. All calculations are executed on computational software MATHEMATICA 11.0

Numerical measures
In order to demonstrate the proposed methodology, we consider four different practical data where denotes the number of classes, = ( ), the ′ being the ordered observations respectively.  Kolmogrov-Smirnov (K-S) statistics are computed to compare the fitted models. The statistics * and * are described in details in [8]. In general, the smaller values of these statistics, better fit to the data. The required computations are carried out in the Mathematica 11.0.

Concluding Remarks:
There has been an increased interest in defining new generated classes of univariate continuous distributions. The extended distributions have attracted several statisticians to develop new models. In this paper we propose the new log Dagum-X family of distributions. We study some of its mathematical properties. The maximum likelihood method is employed for estimating the model parameters. One special model, the distribution Weibull is considered and its properties are studied. It is fitted to two real data sets. The proposed special model consistently better fit than other competing models. We hope that the new family and its generated models will attract wider application in several areas such as engineering, survival and lifetime data, hydrology, economy.