The Third Order Approximation for the Coverage Probability of a Confidence Set Centered at the Positive Part James-Stein Estimator

Abstract

In this paper, we continue the work of Ahmed et al. (2006, 2009, 2015) by investigating the asymptotic expansion approximation for the coverage probability of a confidence set centered at the positive-part James-Stein estimator. The third order Taylor expansion is the main tool here. The theoretical part provides a formula of the approximation for the coverage probability in the case of a noncentrality parameter   where   is the sample size and   is the mean vector of the variate normal distribution with independent components and equal unit variances. In the computational part, we compare the first, second and third orders of the asymptotic expansion with the exact values of the coverage probabilities in order to obtain the accuracy of estimation. The results show that all of these approximations are reliable. However, the first order of the asymptotic expansion gives the best result, especially when the noncentrality parameter  is far from 0.