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 estimatorγ B,θ when θ is known and the Bayes estimator γ B,θ ML plugging the maximum likelihood estimator (MLE)θML in θ ofγ B,θ when θ is unknown are derived. The second order asymptotic loss ofγ B,θ ML relative toγ B,θ is also obtained through their asymptotic variances. Further, it is shown thatγ B,θ andγ B,θ ML are second order asymptotically equivalent to the bias-adjusted MLEŝ γ ML * ,θ andγML * when θ is known and when θ is unknown, respectively. Some examples are also given.

Information-Geometric Approach for a One-Sided Truncated Exponential Family

“…Moreover, whether θ is known does not affect the first-order term of V θ [22]. This is also true for the estimation of θ [23]. Based on these facts, we assume that ∂ i and ∂ γ are orthogonal, and define…”