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

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 τ2, 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 τ2 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.