Estimating the parameter of the population selected from discrete exponential family

Abstract

Let X-1, ..., X-p be p independent random observations, where X-i is from the ith discrete population with density of the form u(i)(theta(i))t(i)(x(i))theta(xi)(i), where theta(i) is the positive unknown parameter. Let X-(1) = ... = X-(l) > X(l+1) >= ... >= X-(m) > X(m+1) = ... = X-(p) denote the ordered observations, where the ordering is done from the largest to the smallest and from smaller index to larger ones, among equal observations. Suppose the population corresponding to X-(1) (or X(m+1)) is selected, and theta((i)) denotes the parameter associated with X-(i), 1 <= i <= p. In this paper, we consider the estimation of theta((1)) (or theta((m+1))) under the loss L-k (t, theta) = (t - theta)(2)/theta(k), for k >= 0, an integer. We construct explicit estimators, specifically for the cases k = 0 and k = 1, of theta((1)) and theta((m+1)) that dominate the natural estimators, by solving certain difference inequalities. In particular, improved estimators for the selected Poisson and negative binomial distributions are also presented. (c) 2007