Estimating the parameter of the population selected from discrete exponential family

Let X1,…,XpX1,…,Xp be pp independent random observations, where XiXi is from the iith discrete population with density of the form ui(θi)ti(xi)θixiui(θi)ti(xi)θixi, where θiθi is the positive unknown parameter. Let X(1)=⋯=X(ℓ)>X(ℓ+1)≥⋯≥X(m)>X(m+1)=⋯=X(p)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)X(1) (or X(m+1)X(m+1)) is selected, and θ(i)θ(i) denotes the parameter associated with X(i),1≤i≤p. In this paper, we consider the estimation of θ(1)θ(1) (or θ(m+1)θ(m+1)) under the loss Lk(t,θ)=(t−θ)2/θkLk(t,θ)=(t−θ)2/θk, for k≥0k≥0, an integer. We construct explicit estimators, specifically for the cases k=0k=0 and k=1k=1, of θ(1)θ(1) and θ(m+1)θ(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.