Bayesian estimation of 3-component mixture of Gumbel type-II distributions under non-informative and informative priors

information is available, elicitation of hyperparameters is given. In order to numerically study the execution of the Bayes estimators under different loss functions, their statistical test termination times. The comparisons among the estimators have been made in terms of the corresponding posterior risks. A real life data example is also given to illustrate the study.


INTRODUCTION
predicting the chance of meteorological phenomena, disasters.It has also been used in describing the life distribution is used to analyse the variables such as monthly, quarterly and annual maximum values of daily the log-Weibull distribution and the double exponential distribution (also considered as the Laplace distribution).Motivated by the above mentioned applications of mixture models, the authors have planned to conduct Bayesian analysis of a 3-component mixture of the proportions.The parameters of component distributions three different loss functions are used for the Bayesian analysis.In addition, an ordinary type-I right censored sampling scheme is also assumed.

3-component mixture of Gumbel type-II distributions
The probability density function (PDF) and the type-II distribution for a random variable x are given by: , 0 , 0 , exp , ; The likelihood function Suppose 'n' units from the 3-component mixture of According to Mendenhall and Hader (1958), there are can be pointed out easily as a subset of subpopulation-I, subpopulation-II or subpopulation-III.Out of 'r' units, suppose r 1 , r 2 and r 3 units belong to subpopulation-I, subpopulation-II or subpopulation-III, respectively and such that r = r 1 + r 2 + r 3 x lk x lk < t be the failure time of k th unit belonging to the l th subpopulation, l = 1,2,3 and k = 1,2,...,r l .For a 3-component ... (7) given by: .

Posterior distribution using the non-informative and the informative priors (IP)
In this section, posterior distributions of parameters given data, say x, are derived using the non-informative (uniform and Jeffreys) and the informative (gamma) priors.

Posterior distribution using uniform prior (UP)
or little prior information is given, usually the noninformative prior is assumed to be the UP.UPs over the ( ) The joint posterior distribution of parameters 1 , 2 , 3 , p 1 and p 2 given data x assuming the UP is: ( )

Posterior distribution using the Jeffreys prior (JP)
According to Jeffreys (1946;1961) and Berger (1985), , where , where is the Fisher's information matrix.The pri is the Fisher's information matrix.
The prior distributions of the mixing proportions p 1 and p 2 are again taken to be uniform over the interval (0,1).The joint prior distribution of parameters The joint posterior distribution of parameters ,

Posterior distribution using the gamma prior (GP)
prior for the component parameters is given by: The joint posterior distribution of parameters    .

Bayes estimators and posterior risks using the UP, JP and IP under SELF
The squared error loss function (SELF) has been suggested So, the Bayes estimators and posterior risks using the UP, JP and IP for parameters Where 1 for the uniform prior, 2 for the Jeffreys prior and 3 for the gamma prior.

Bayes estimators and posterior risks using the UP, JP and IP under precautionary loss function (PLF)
Norstrøm (1996) discussed an asymmetric precautionary loss function (PLF) and also introduced a special case of The Bayes estimator and posterior risk are: , respectively.The respective marginal posterior distribution yields the Bayes estimators and posterior risk using the UP, JP and IP for parameters p and 2 p under PLF as: , respectively.The Bayes estimators and posterior risks using the UP, JP and IP for parameters p under DLF are:

Bayes estimators and posterior risks using the UP, the JP and IP under DeGroot loss function (DLF)
Journal of the National Science Foundation of Sri Lanka 45(3)

Elicitation of hyperparameters
Elicitation is the key task for subjective Bayesian.The elicitation.Aslam (2003) proposed different methods of elicitation based on prior predictive distribution for the method of elicitation using prior predictive distribution based on predictive probabilities.In this method, the particular intervals of the random variables.The set the elicited probabilities and the expert predictive probabilities is minimum, is considered.

Elicitation of hyperparameters using the gamma prior
For eliciting the hyperparameters, prior predictive distribution (PPD) is used.The PPD for a random variable y is: We choose the prior predictive probabilities satisfying the prior density.Using the prior predictive distribution, 0.67, 0.13, 0.06, 0.03, 0.02, 0.01, 0.01, 0.008 and 0.006, respectively given an expert opinion.
For eliciting the hyperparameters a 1 , a 2 , a 3 , b 1 , b 2 , b 3 , a, b and c, the equations are simultaneously solved through the computer programme developed in SAS package using the 'PROC SYSLIN' command; the values of the hyperparameters are found to be 1.7388, 0.4722, 0.3528, 1.0779, 0.1469, 0.0737, 0.3695, 1.4360 and 8.8749, respectively.

Limiting expressions
Letting t , all the observations that are incorporated in our analysis are uncensored and therefore, r tends to n, r 1 n 1 , r 2 n 2 and r 3 n 3 .As a result, the amount of consequently results in the reduction of the variances of the estimates.The limiting (complete sample) expressions for Bayes estimators and posterior risks using the UP, JP and IP under SELF, PLF and DLF are and their corresponding posterior risks using complete sampling under SELF are given in Table VII in appendix.

RESULTS AND DISCUSSION
In order to obtain and investigate the properties of the derived estimators, simulation analysis has been carried illustrate the properties of the Bayes estimators of the by using different values of the vector of the parameters = {(2, 3, 4, 0.20, 0.60), (3, 3, 3, 0.40, 0.40)}.
set of parameters, the observations are randomly taken declared as censored observations.For each t, only failures I, subpopulation II or subpopulation III.This process programme.For each of the 1000 samples, the Bayes results are presented in the appendix from Tables I to VII.
The simulation study provides some interesting properties of the Bayes estimates.The properties are highlighted in parameters, different loss functions and censoring rates.
based on simulation analysis tables corresponding to the different prior distributions and various loss functions.In colours, respectively.In Table VII in appendix, some results for the complete the importance of incorporating full information into the x Bayes estimates and Bayes posterior risks (BPRs) using in Table 5.
It is noted that the results gained from real data are that the execution of the informative prior is better than the non-informative priors.It is also examined that execution of DLF is preferred for estimating estimating the proportion parameters.

CONCLUSION
As far as the problem of selecting the most suitable prior is concerned, it can be seen that the informative prior other hand, DLF is observed performing better than PLF for estimating proportion parameters, SELF is observed superior to PLF and DLF.Therefore, the informative prior has a clear edge over the non-informative prior.
doubly censored samples by considering various loss functions.Furthermore, Salinas et al. (2012) proposed type-II right censored data.Abbas et al. (2013) obtained type-II distribution under different loss functions and the maximum likelihood method.Reyad and Ahmed based on type-II censored samples.These estimators are obtained under different symmetric and asymmetric loss functions.Mixture models play an important role in many mixture of some suitable probability distributions is recommended to study a population that is supposed to different parameters of mixture distributions.Chen et al. (1989) considered the Bayes estimation for mixtures Saleem et al. (2010) presented the Bayesian analysis of et al. (2012) developed the Bayesian analysis of 2-component September 2017 Journal of the National Science Foundation of Sri Lanka 45(3) and Aslam (2013) discussed the Bayesian estimation under informative priors using different loss functions.Noor and Aslam (2013) studied Bayesian inference of the inverse Weibull mixture model under type-I censoring, and Ali (2015) described mixture of inverse Rayleigh One particular feature, often present in time-to-event of censoring such as right censoring, left censoring and interval censoring, single or multiple censoring, and type-I and type-II censoring.Type-I and type-II censoring schemes are most familiar among them.
estimator and posterior risk under SELF are:

Table 1 :
Limiting expressions for the Bayes estimators as t September 2017Journal of the National Science Foundation of Sri Lanka 45(3)

Table 2 :
Limiting expressions for the posterior risks as t

Table 3 :
Limiting expressions for the posterior risks as t

Table 4 :
Limiting expressions for the posterior risks as t

Table VII :
Bayes estimates and posterior risks (in parentheses) under SELF using complete sampling