Bayesian Analysis of the Mixture of Frechet Distribution under Different Loss Functions

This paper has to do with 3-component mixture of the Frechet distributions when the shape parameter is known under Bayesian view point. The type-I right censored sampling scheme is considered due to its extensive use in reliability theory and survival analysis. Taking different non-informative and informative priors, Bayes estimates of the parameter of the mixture model along with their posterior risks are derived under squared error loss function, precautionary loss function and DeGroot loss function. In case, no or little prior information is available, elicitation of hyper parameters is given. In order to study numerically, the execution of the Bayes estimators under different loss functions, their statistical properties have been simulated for different sample sizes and test termination times. A real life data example is also given to illustrate the study.


Introduction
Frechet distribution was introduced by a French mathematician named Maurice Frechet (1878Frechet ( , 1973) who had determined before one possible limit distribution for the largest order statistic in 1927. The Frechet distribution has been manifested to be helpful for modeling and analysis of several extreme events ranging from accelerated life testing to earthquakes, floods, rain fall, sea currents and wind speed.
Applications of the Frechet distribution in many fields given in Harlow (2002) showed that it is an important distribution for modeling the statistical behavior of materials properties for a variety of engineering implementation. In hydrology, the Frechet distribution is applied to extreme events such as annually maximum one day rainfalls and river discharges. Nadarajah and Kotz (2008)  The structure of this article is as follows. The Frechet mixture model along with its likelihood function is formulated in section 2. The expressions for posterior distributions using the non-informative and informative priors are derived in section 3. In section 4, the Bayes estimators and posterior risks using the uniform, the Jeffreys', the exponential and the inverse levy priors under squared error loss function (SELF), precautionary loss function (PLF) and DeGroot loss function (DLF) are presented. The elicitation of hyperparameters is given in section 5. In section 6, the limiting expressions of the Bayes estimators and their posterior risks are derived. The simulation study and the real data applications are presented in section 7 and 8, respectively. This article concludes with a brief discussion in section 9.

3-Component mixture of the Frechet distributions
The probability density function (p.d.f) and the cumulative distribution function (c.d.f) of the Frechet distribution for a random variable X are given by: (1) Where the parameter α > 0determines the shape of the distribution and β > 0is the scale parameter.

The posterior distribution using the Jeffreys' prior (JP).
According to Jeffreys' (1946Jeffreys' ( , 1998 and Berger (1985), the JP is defined as is the Fisher's information matrix. The prior distributions of the mixing proportions 1 p and 2 p are again taken to be the uniform over the interval   The joint posterior distribution of parameters β1, β2, β3, p1 and p2 given data x assuming the JP is:

Bayes estimators and posterior risks using the UP, the JP, the Exponential and Inverse Levy prior under SELF, PLF and DLF
If dˆ is a Bayes estimator, then   d  is called posterior risk and is defined as: Our purpose, in this study, is to look for efficient Bayes estimates of the different parameters. For this purpose, three different loss functions, namely SELF, PLF and DLF used to obtain Bayes estimators and their posterior risks. The SELF, defined as ,was introduced by Legendre to develop the least squares theory. Norstrom (1996) discussed an asymmetric PLF and also introduced a special case of general class of PLFs, which is defined as For a given prior, the Bayes estimator and posterior risk under SELF are calculated as: ,respectively. Similarly, the Bayes estimators and posterior risks with PLF and DLF are given by: where v=1 for the UP, v=2 for the JP, v=3 for the EP and v=4 for the ILP.

The
Bayes estimators and posterior risks using the UP, the JP and IP under PLF. Norstrom (1996) discussed an asymmetric PLF and also introduced a special case of general class of PLFs, which is defined as . The Bayes estimator and posterior risk are: , respectively. The respective marginal posterior distribution yields the Bayes estimators and posterior risk using the UP, the JP and the IP for parameters

Elicitation of Hyper-parameters
Elicitation is the main task for subjective Bayesian. The complete procedure for quantifying the prior information in the form of prior distribution is precisely known as the elicitation. Aslam (2003) presented different methods of elicitation based on prior predictive distribution for the elicitation of the hyper-parameters. In this study, we use the method of elicitation using prior predictive distribution based on the predictive probabilities. In this approach, confidence levels of the prior predictive are gained for the particular intervals of the random variables. The set of hyper parameters, for which the difference between the elicited probabilities and the expert predictive probabilities is minimum, is considered.

Elicitation of hyper-parameters using the Exponential Prior.
For eliciting the hyper-parameters, prior predictive distribution (PPD) is used. The PPD for a random variable X is: We choose the prior predictive probabilities, satisfying the laws of probability, to elicit the hyper-parameters of the prior density. Using the prior predictive distribution, we consider the six intervals (0,1), (1,2), (2,3), (3,4), (4,5) and (5,6) with probabilities 0.73, 0.11, 0.05, 0.02, 0.02, and 0.01 respectively, given an expert opinion. The following nine equations are derived from the above information: For eliciting the hyper parameters k1, k2, k3, a, b and c and the equations are simultaneously solved through the computer program developed in SAS package using the 'PROC SYSLIN' command, the values of the hyper parameters are found to be 2.0003,3.0030,4.0016,2.0103,1.7607and 1.50 respectively.

Elicitation of hyper parameters using the Inverse Levy
Prior. The PPD using Inverse Levy prior for a random variable X is given by: Using same canon defined as above for the exponential prior, the values of the hyperparameters a1, a2, a3, a, b and c are 1.9520, 2.5321, 3.7735, 0.2763, 0.1167 and 1.0.

Limiting Expressions.
Letting t → ∞, all the observations that are assimilated in our analysis are uncensored and therefore r tends n, r1tends to the unknown n1, r2tends to the unknown n2 and r3tends to the unknown n3. As a result, the amount of information carried in the sample expands, which results in the depletion of the variances of the estimates.
The limiting (complete sample) expressions for Bayes estimators and posterior risks using the UP, the JP, the EP and the ILP under SELF, PLF and DLF are given in the Tables 1-6.

Simulation Study
A comprehensive simulation study was conducted in order to explore the performance of the Bayes estimators, impact of sample size and censoring rate to be appropriate for the model. Samples of sizes n=25, 40, 55 are generated from a 3-component mixture of the Frechet distributions with various set of the parametric values β1, β2, β3, p1 and p2 fixed as (β1, β2, β3, p1, p2) = (0.50, 1.0, 1.50, 0.30, 0.50), (1.50, 1.0, 0.50, 0.50, 0.30). For fixed sample size, test termination time and set of parameters, the simulation is repeated 1000 times and the results are then averaged. Sample of sizes p1n, p2n and (1 − p1 − p2) n are chosen randomly from first component densityf1 (x; θ1), second component density f2 (x; θ2) and third component densityf3 (x; θ3), respectively. The observations which are greater than a fixed t are declared as censored observations. For each t only failures have been inspected either as a member of subpopulation-I or subpopulation-II or subpopulation-III. On the basis of each sample size, the Bayes estimators (BEs) and Posterior risks (PRs) are computed using the informative and noninformative priors under SELF, PLF and DLF. In order to conduct Bayesian analysis under informative priors, elicitation of hyper-parameters is obtained by using the prior predictive approach. In order to evaluate the impact of test termination time on Bayes estimators, the Type-I right censoring scheme is used for fixed test termination time t=15 and 20. For each of the 1000 samples, the Bayes estimators and Posterior risks were calculated using a routine in Mathematica 10.0 and the results are presented in Tables 9-16. The simulation study gives us some interesting characteristics of the Bayes estimates. The properties have been foregrounded in terms of sample sizes, size of mixing proportion parameters, different loss functions and censoring rates. It is noticed that because of censoring, the posterior risks of all the parameters are reduced with an increase in sample size.  Bayes estimates and Bayes Posterior risks using the UP, the JP, the EP and the ILP under SELF, PLF and DLF are in Table 17 given in appendix.

A Real Life Data Application
It is noted that the results gained from real data are compatible with simulation results. The results declare that the execution of the informative prior is better than the noninformative priors. It is also examined that execution of DLF preferred for estimating the component parameters, while SELF better for estimating the proportion parameters.

Final Remarks
In this study, the Bayesian estimation of 3-component mixture of the Frechet distributions has been considered assuming the case when the shape parameter is known based on type-I censored data. The purpose of this paper is to find out the appropriate combinations of prior distributions and loss functions to estimate the parameters of the 3component mixture of the Frechet distributions. We conducted all-encompassing simulation study to find out the relative performance of the Bayes estimators when the shape parameter is assumed to be known. From simulated results, we observed that an increase in the sample size and test termination time provides better Bayes estimators. Furthermore, as sample size increases (decreases) the posterior risks of Bayes estimators' decreases (increases) for a fixed test termination time. Also, the DLF is perceived as an appropriate choice for estimating component parameters and SELF is expedient for estimating the proportion parameters. Finally, we deduce that the EP is apt prior in order to estimate the component parameters. When SELF is used, the EP is an appropriate prior for proportion parameters. The similar pattern is examined for the JP when noninformative priors are contemplated.