Mixture Modeling of Exponentiated Pareto Distribution in Bayesian Framework With Applications of Wind-Speed and Tensile Strength of Carbon Fiber

Mixture modelling has stunning applications to explain the composite problems in simple way. Bayesian demonstration of 3-Component mixture model of Exponentiated Pareto distribution in right-type-I censoring scheme is presented in this article. The posterior densities of the parameter(s) are attained supposing the non-informative (uniform, Jeffreys) priors. The symmetric and asymmetric Loss Functions (Squared Error, Precautionary, Quadratic and DeGroot Loss Function) are assumed to get the Bayes estimator(s) and posterior risk(s). The presentation of the Bayes estimator(s) over posterior risk(s) in the studied loss functions is examined over simulation practice. Two real-data sets, wind speed and tensile strength of carbon fiber, are also analyzed for mixture to complete the performance of Bayes estimator(s). To enhance the study, the limiting forms are also derived for Bayes estimator(s) and posterior risk(s). The results reveal that for the component parameter(s), the Bayes Estimator(s) have their risks accordingly: DeGroot Loss Function < Precautionary Loss Function < Squared Error Loss Function < Quadratic Loss Function, and whereas for the proportion parameter(s) these are classified as: Squared Error Loss Function < Precautionary Loss Function < DeGroot Loss Function < Quadratic Loss Function. Therefore, in this study, DeGroot Loss Function performs efficient and the most preferable non-informative prior is the Jefferys prior for estimation of 3-Component mixture of Pareto distribution.


I. INTRODUCTION
The Pareto distribution, is the power law probability density and extensively used in geographical, social, actuarial, scientific and many other areas. To model the earthquakes and forest fire areas [1] used Pareto distribution. Reference [2] discussed applications of Pareto distribution to estimate the model for disk drive errors. Pareto distribution extended as beta Pareto studied by [3], Kuaraswamy Pareto explored The associate editor coordinating the review of this manuscript and approving it for publication was Chaitanya U. Kshirsagar. by [4], beta exponentiated Pareto introduced by [5], Gamma Pareto presented by [6]. Reference [7] introduced an exponentiated Weibull Pareto distribution and discussed its several characteristics comprising reliability and hazard function. Reference [8], used exponentiated Weibull distribution to model the bathtub-data. Reference [9] studied Weibull Pareto distribution with its applications. Reference [10] suggested new Weibull Pareto distribution. Reference [11] explored exponentiated Pareto distribution. Reference [12] studied the behavior of different methods of parameters estimation of exponentiated Pareto distribution. Reference [13] explored mixture of exponentiated Pareto and exponential model under type-II censoring scheme.
Mixture modelling is widely used in many situations, particularly whenever we have more than one sub population. The number of component in a mixture modeling is in line for to heterogonous property of the parental population. It is therefore limited to be finite, while in several cases the elements may be infinite. As compare to simple composition, it delivers more appropriate explanation of many analytical frame works. Mixture modeling has extensive uses in survival analysis. The 3-Component mixture is important than 2-Component mixture in the context to fix and determine the fault of more than 2-Components. Already in literature Bayesian mixture work on many distributions is done [see: [14][15][16]. Recently, [17] introduced 3-Component mixture model of exponentiated Weibull distribution among Bayesian paradigm. Moreover, [18] discussed 3-Component mixture for Pareto distribution by using right type-I censoring pattern. Reference [19] explored 3-Component scheme under Bayesian approach for exponentiated inverted Weibull distribution. We considered 3-Component mixture modelling of exponentiated Pareto distribution in right type-I censoring procedure using non-informative priors in this study.
Encouraged by the above stated researches of 3-Component mixture, we examine the Bayesian analysis mixture of 3-Component model of EPD in this article. The key attention of this study is to list the competent Bayes Estimators (BEs) of Component and Proportion parameter(s). For the sack, two asymmetric and two symmetric LFs are used with (Non-Informative Priors) NIPs, UP (Uniform Prior) and JP (Jeffreys Prior) to achieve such results. The estimator(s) are explored under the type-I right censoring procedure. The study algorithm is illustrated in the following FIGURE 1.
The remaining study is intended as bellows. The 3-Component mixture model of EPD is designed in the next section, with the suggested BEs of several parameter(s) assuming different LFs among the limiting expressions. Simulation and Real data applications are elaborated in section III. To finish in section IV conclusions are provided.

A. THE 3-COMPONENT MIXTURE OF EPD
For a random variable X , the pdf (probability density function) for the shape parameter with the cdf (cumulative distribution function) of EPD can be illustrated as: Here for EPD the shape parameter is denoted by α i . With the w 1 and w 2 mixing fraction, a defined 3-Component mixture model can be stated as: For mixing proportion parameter(s) and some component points, a 3-Component mixture of the EPD is displayed in the following FIGURE 2. For 3-Component mixture the cdf is stated as:

B. THE POSTERIOR DISTRIBUTION USING THE NON-INFORMATIVE PRIORS
The prior evidence has main part to differentiate the Bayesian and Classical study. The prior is defined as the characterization of the uncertainty about the parameter, existing to the prior knowledge. Prior distribution categorized into two VOLUME 8, 2020 classes as non-informative and informative. A prior distribution differentiated as NIP, if it is even relative to likelihood function. Whereas, an IP (Informative Prior) is distinct as a P (prior) which has a link concerning the posterior distribution and is not the result by the probability function. In this segment, posterior distributions through the likelihood are estimated under the NIPs. Assuming that from the 3-Component mixture modelling of EPD n units are obsessive in a life testing process with fixed (test termination time) t. Assume that the designated outcome exposed that n components from r failed till t fixed and the n-r which are remaining units are still at in running stage. Here important to note that because of the failures, from r, r 1 categorized as related to subpopulation-I, r 2 belongs to subpopulation-II and r 3 are connected to subpopulation-III. The total uncensored sample elements are categorized as r = r 1 , r 2 and r 3 . Whereas, the rest of the n-r sample points are consider as censored. Now we have distinct the time-to-failure, of the ith-unit connecting to lth sub-population as x oi , 0 < X oi ≤ t, where l = 1, . . . , 3; and i = 1, . . . , r l .
The likelihood of the 3-Component model can be defined as: After solving the above term, we get likelihood of 3-Component mixture of EPD as: where The widely used NIPS includes UP and JP. Most of the researchers cited that UP is the furthermost studied prior, for the assessment, of unknown parameter(s) of concern (see [20]- [22]). The improper UP is considered for the component parameter(s) of EPD as: α 1 , α 2 and α 3 , i.e., Moreover, the UP in the interval (0, 1) is used for the parameter(s) w 1 and w 2 , i.e., w 1 ∼ U (0, 1) and w 2 ∼ U (0, 1). The joint prior density of parameter(s) α 1 , α 2 , α 3 , w 1 and w 2 can be stated as: π 1 ( ) = 1; α 1 , α 2 and α 3 > 0; w 1 and w 2 ≥ 0; w 1 + w 2 ≤ 1 So, for the UP the joint posterior distribution of parameter(s) α 1 , α 2 , α 3 , w 1 , w 2 can be written as: where, 31 31 and B (A 01 , B 01 , C 01 ) is the Beta function and is expended as The Jeffreys suggested a rule of thumb to demonstrate NIP for parameter θ as: θ , under transformation of the variables (see [23]- [26]). While, for proportion parameter(s) w 1 and w 2 the JP is classified as w 1 ∼ U (0, 1) and w 2 ∼ U (0, 1). The joint prior density of the parameter(s), under the theory of independence of the studied parameter(s) α 1 , α 2 , α 3 , w 1 and w 2 is specified as: Then the joint posterior distribution turns out to be: where, The graphs of marginal posterior distribution of component parameter(s) are depicted in the following FIGURES 3(a-c). FIGURES 3(a-c), represent graphs of the marginal posterior densities of component parameter(s) assuming the NIP for the 3-component mixture of EPD. From FIGURES 3(a-c) it is observed that, the graphs of marginal posterior densities for the shape parameter of the mixture of EPD for component parameter(s) indicates symmetrical pattern with slight variation for both t. The graph of the first component tends to be more peaked than the second and third component(s).

C. BAYES ESTIMATORS AND POSTERIOR RISKS UNDER LOSS FUNCTIONS
The existent valued function which represents estimator loss among the particular value of parameter is identified as LF (Loss Function). The present unit deliberated, BEs and PRs over four different symmetric and asymmetric LFs, i.e.; Squared Error Loss Function (SELF), Precautionary Loss Function (PLF), Quadratic Loss Function (QLF) and DeGroot Loss Function (DLF). Here, we discuss theses LFs for 3-Component mixture of EPD one by one.
For d, PR ρ (d) can be written as: We get BEs and PRs supposing NIPs for the component as well as for proportion parameter(s) α 1 , α 2 , α 3 , w 1 and w 2 under SELF as:  (20) where v = 1 for the UP and for the JP v = 2. QLF is classified as symmetric LF and have applications in least square scheme. It desires additional attention due to its property of variance and symmetric. Where L (β, d) = α (β − d) 2 is the QLF. We express the BE and the PR among QLF as: By utilizing this idea under UP and JP the derivation of BEs and PRs is (21)- (30), as shown at the bottom of the page, where v = 1 for the UP and v = 2 for the JP. The next, studied LF is the PLF an asymmetric LF, which first explored by [27]. The simplified form of PLFs is a special case expressed as: Under PLF the results of BEs and PRs are derived by: The resulting BEs and PRs under the studied Ps and LF are obtained as: where v = 1 for the UP and for the JP v = 2.
The fourth used LF in this study is another mostly applied asymmetric LF explored by [28] is DLF, described as: Under DLF the BEs and PRs may . The obtained BEs and PRs expending the studied Ps are: where v = 1 for the UP and v = 2 for the JP.

D. LIMITING EXPRESSIONS
In contents of uncensored sampling technique limiting terms have extensive uses. When test-termination-time t → ∞, it may be noted that uncensored outcomes r approaches to n (sample size) and r l approaches to n l , where l = 1, . . . , 3. The censored elements become to be uncensored and also, the information mentions in the sample have also increased here. In end result, the efficiency of the BEs is also increases rapidly for to the contribution of all the values in sample. Therefore, limiting terms for the NIPs is easily estimated.
The limiting derivations assuming UP and JP for the BEs and PRs of 3-Component exponentiated Pareto distribution are reported in Table 1.

A. SIMULATION BASED RESULTS
Simulation is organized to evaluate the performance of BEs under UP and JP and with some symmetric and asymmetric LFs for the 3-Component mixture of the EPD. The computational code of simulation analysis is mention in Appendix.
• The first step, we need to create few sample sizes n = 50, 100, 200, 500 supposing some parameter values fixed as • Outcomes are averaged with the help of Mathematica software, by replication of the simulation process 1000 intervals.
• Randomly selected sample-sizes w 1 n, w 2 n and (1 − w 1 − w 2 ) n are belongs to I, II, and the III factor densities, respectively.
• To determine the effect of t on BEs the right-type-I censoring technique, is used.
• All points are known as censored which were greater than t.
• Two fixed censoring times t are applied to measure the behaviour of censoring rate upon the estimates.
• In favour of the fixed t, values are as: t = 25 and 30.
• The point(s) which are greater than t are considered as the censored observations.
From Table 2, it is clear that BEs assuming all stated LFs and NIPs is greater for the lesser n over to the greater n for both t. It is also noted, for both t the variation of the BEs from supposed components near to zero with the increase in n. Though, the PRs supposing the stated priors and LFs decrease with the raise in n. FIGURE 4(a) explains the performance of BEs and indicates that for both t BEs performs best. FIGURE 4(b) illustrate the PRs and depicts that DLF have minimum PR and consider best LF among the other LFs for 3-Component mixture of EPD. From FIGURE 4(a-b) it is also clear that DLF have minimum risk for the component parameter(s). In favors of the valuation of component measure of the parameter(s) PRs gives smaller outcomes among the DLF; over to the results revealed under the SELF, QLF and PLF at different n and t. For proportion parameter(s) estimation, it is described that SELF illustrates minimum PRs among QLF, PLF and at last DLF. Therefore, results direct that JP is the most suitable prior than the UP for present study. In view of simulation results under the mentioned LFs; the DLF is considered best for assessing the component parameter(s) and SELF observed efficient for proportion parameter(s).
Neutrosophic statistics is the generalization of classical statistics. The current study can be extended using neutrosophic statistics [34]- [39].

C. COMPARISON
From TABLE 2-3, it is distinguished that simulation analysis are compatible with both (Wind Speed and Carbon Fiber) real data sets. There seem few concessions which are due to the causes of the small data set. In both, simulation as well as real data applications, outputs under the JP are most precise as compare to the UP. Though, DLF as compare to other studied LFs (SELF, QLF, PLF) demonstrates better results for the proportion segment of parameter(s).

IV. CONCLUSION
In this article, the Bayesian construction mixture of the 3-Component model of EPD under right type-I censoring procedure is considered. The comprehensive simulation technique is formulated to evaluate and establish some significant characteristics about the BEs of the 3-Component mixture of EPD supposing the NIPs under the four symmetric and asymmetric LFs. Over-estimation and under-estimation of proportional mixture are inversely connected to the sample size and are directly relative to censoring percentage. A lesser sample size, a huge censoring degree reasons the higher level of over-estimation. But this consequence may be condensed by assuming a large sample size. Posterior densities are estimated and reported that they are in closed forms. The other objective of this article was the choice of suitable LF and P for the conclusion of mixture parameter(s) at various n and t. To conclude the performance, we estimated several posterior quantities, like BEs with their PRs by considering diverse n and t. The limiting formulation for the BEs and PRs are also derived under the studied LFs (SELF, DLF, QLF, PLF) among the NIPs (JP, UP). The well-matched outcomes are noted for simulation as well as for real data observations study, to investigate the presentation of BEs. Figure 4 also demonstrate the performance of BEs and PRs and conclude that BEs and LFs are greater for small sample size for both t as compare to large sample sizes. While, the PRs using the NIPs and LFs are decreased by increasing the sample size for both t. The contact of several n and t are estimated for BEs. The outcomes revealed that, for the component parts the direction of best BEs are as DLF < PLF < SELF < QLF, and on other hand for the proportion segment analysis this classifying is noticed: SELF < PLF < DLF < QLF. Therefore, we conclude that the efficient and most suitable P is JP over the UP due to the minimum risk. DLF perform efficient for 3-Component mixture model of EPD among the PLF, SELF and QLF.