Second Order Asymptotic Variance of the Bayes Estimator of a Truncation Parameter for a One-Sided Truncated Exponential Family of Distributions

Abstract

For a one-sided truncated exponential family of distributions with a truncation parameter γ and a natural parameter θ as a nuisance parameter, the stochastic expansions of the Bayes estimator ˆ γ B,θ when θ is known and the Bayes estimator ˆ γ B, ˆ θ ML plugging the maximum likelihood estimator (MLE) ˆ θ ML in θ of ˆ γ B,θ when θ is unknown are derived. The second order asymptotic loss of ˆ γ B, ˆ θ ML relative to ˆ γ B,θ is also obtained through their asymptotic variances. Further, it is shown that ˆ γ B,θ and ˆ γ B, ˆ θ ML are second order asymptotically equivalent to the bias-adjusted MLEs ˆ γ ML ∗ ,θ and ˆ γ ML ∗ when θ is known and when θ is unknown, respectively. Some examples are also given.