High order approximation for the coverage probability by a confident set centered at the positive-part James–Stein estimator

Abstract

In this paper we continue our investigation connected with the new approach developed in Ahmed et al. [Ahmed, S.E., Saleh, A.K.Md.E., Volodin, A., Volodin, I., 2006. Asymptotic expansion of the coverage probability of James–Stein estimators. Theory Probab. Appl. 51 (4) 1–14] for asymptotic expansion construction of coverage probabilities, for confidence sets centered at James–Stein and positive-part James–Stein estimators. The coverage probabilities for these confidence sets depend on the noncentrality parameter τ 2 , the same as the risks of these estimators. In this paper we consider only the confidence set centered at the positive-part James–Stein estimator. As is shown in the above-mentioned reference, the new approach provides a method to obtain for the given confidence set, an asymptotic expansion of the coverage probability as one formula for both cases τ → 0 and τ → ∞ . We obtain the third terms of the asymptotic expansion for both mentioned cases, that is, the coefficients at τ 2 and τ − 2 . Numerical illustrations show that the third term has only a small influence on the accuracy of the asymptotic estimation of coverage probability.