Estimating the parameter of the population selected from discrete exponential family

P. Vellaisamy and Sushmita Jain

Statistics & Probability Letters, 2008, vol. 78, issue 9, 1076-1087

Abstract: Let X1,...,Xp be p independent random observations, where Xi is from the ith discrete population with density of the form ui([theta]i)ti(xi)[theta]ixi, 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 . In this paper, we consider the estimation of [theta](1) (or [theta](m+1)) under the loss Lk(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.

Date: 2008
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