IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 12, Issue 2 Ver. V (Mar. - Apr. 2016), PP 25-35 www.iosrjournals.org DOI: 10.9790/5728-1202052535 www.iosrjournals.org 25 | Page Estimation of a Mixture of Two Weibull Distributions under Generalized Order Statistics Neamat S. Qutb 1 , Samia A. Adham 2 ,Nojoud K. Dandeni 3 1,2,3 (Department Of Statistics, Faculty Of Science/ King Abdulaziz University, Saudi Arabia.) Abstract:This paper deals with the estimation of the parameters, reliability and hazard rate functions of the mixture of two Weibull distributions (MTWD), with a common shape parameter, based on the generalized order statistics (GOS). The maximum likelihood and Bayes methods of estimation are used for this purpose with standard errors and credible intervals. The Markov Chain Monte Carlo (MCMC) method is used for obtaining Bayes estimates under the squared error loss function. Our results are specialized to progressive Type II censoring and Type II censoring. Comparisons are made between Bayesian and maximum likelihood estimates and between the two censoring types, progressive Type II censoring and Type II censoring. A real data set is used for illustration purpose. Keywords: Mixture of two Weibull distributions, generalized order statistics (GOS ), Maximum likelihood estimation, Bayesian estimation, Markov Chain Monte Carlo I. Introduction The finite mixture of distributions have multiple uses in a various scientific fields such as physics, biology, medicine and industrial engineering, among others. In fact, in life testing, each failure occurs may not has one type of failure, it could be categorized to more than one type. Moreover, the failure time populations could be heterogeneous since it could be consisting of weak components corresponding to short lives and strong components corresponding to long lives. Mixtures of Weibull distributions are widely used to model lifetime data. That is because of the flexibility of the Weibull distribution in modeling both increasing and decreasing failure rates. This flexibility mainly depends on the shape parameter; which led us, when estimating the distribution parameters, to let the value of this parameter to be known and controlled by the researcher carrying out the application. Where, it could be selected according to the type of failure rate that fits the data. For the increasing failure rate data, the shape parameter can be set to a value greater than one. While, for the decreasing failure rate data, the shape parameter can be set to a valueless than one; but for the constant failure rate data, the value of the parameter will be equal to one. This study is concerned with studying the finite mixture of two Weibull distributions, denoted by MTWD, as a lifetime model with two unknown scale parameters and a common known shape parameter. Some of the most important references that discussed different types of mixtures of distributions are the monographs by [1], [2], [3] and [4]. [5]considered Bayesian estimation of the mixing parameter, mean and reliability function of a mixture of two exponential lifetime distributions based on right censored samples.[6] studied and compared classical and Bayesian estimates of the parameters of a finite mixture of two Gomperetz lifetime models based on simulated data sets with Type I and Type II censoring. [7]obtained Bayesian predictive density of order statistics based on finite mixture models. Based on Type I censored samples from a finite mixture of two truncated Type I generalized logistic components,[8] computed Bayes estimates of parameters, reliability and hazard rate functions.[9]considered estimation for the parameters of mixed exponential distribution based on record statistics.[10]considered Bayes inference under a finite mixture of two compound Gompertz components model. [11]considered estimation forthe parameters of mixture of two component exponentiated gamma distribution. [12]applied the order statisticson a mixture model of exponentiated Rayleigh and exponentiated exponential distributions. [13]estimated the parameters of a two-parameter weighted Lindley distribution based on hybrid censoring.[14]introduced the finite mixture of two exponentiatedKumaraswamy distributions. Mixtures of Weibull distributions are widely used to model lifetime data and they have been considered extensively by many authors, [15] estimated the parameters of a mixture Weibull distribution using MLE and Bayes estimation under Type I censoring. [16]studied the classification of the aging properties of generalized mixtures of two or three Weibull distributions in terms of the mixing weights, scale parameters and a common shape parameter. [17]estimated the parameters from the mixture of two Weibull distributions under the informative and non-informative priors they also determined, the Bayes predictive intervals. A mixture of two and three Weibull distributions were used to analyze the data of failure times [18]. [19]proposed a mixture Weibull proportional hazards model to predict the failure of a mechanical system with multiple failure modes.