Estimation of ordered means and P(X<Y) in the exponential case

Abstract

chapter 2 This thesis is concerned with the problem of estimating parameters and the probability $R = P(X < Y)$ and determining sample sizes when $X$ and $Y$ are independent exponential random variables. with a common location parameter. In estimation problem, uniformly minimum variance estimator of $R$ is obtained and Bayes estimators for parameters and $R$ are also obtained under order restriction on parameters and asymmetric loss function. This thesis is divided into the following four parts. (i) Uniformly minimum variance unbiased estimator of $R$ is obtained for two independent two-parameter exponential random variables with unknown common location parameter. The asymptotic distribution of the maximum likelihood estimator is obtained and then the asymptotic equivalence of the maximum likelihood estimator and uniformly minimum variance unbiased estimator of $R$ is established. Performance of the two estimators for moderate sample sizes is studied by Monte Carlo simulation. An approximate interval estimator is also obtained. (ii) Bayes estimation of parameters and $R$ is considered for two independent exponential random variables $X$ and $Y$ with ordered means and known location parameters. Order restricted Bayes estimators for means and $R$ are obtained with respect to inverted gamma and noninformative prior distributions, and their asymptotic properties are established. These estimators are compared with the corresponding unrestricted Bayes estimators by Monte Carlo simulation. (iii) The problem of Bayes estimation for scale and location parameters of a two-parameter exponential distribution is considered with respect to conjugate and noninformative prior distributions under asymmetric loss function. A reference prior of $R$ is derived and Bayes estimators are also obtained with respect to the conjugate and reference priors under asymmetric loss when the location parameters are known. (iv) The problem of choosing sample size for reliability verification in...