James-Stein confidence set: Equal area approach to the global approximation of coverage probability

Abstract

In [S.A. Ahmed, A.K.MD.E. Saleh, A.I. Volodin, and I.N. Volodin, “Asymptotic Expansion of the Coverage Probability of James-Stein Estimators,” Theory Probab. Appl. 51, 683–695 (2007)], an asymptotic expansion of coverage probabilities for the James-Stein confidence sets was constructed, which was asymptotically exact for both large and small values of the noncentrality parameter τ2, that is, the sum of squares of the means of p ≥ 3 normal distributions subject to confidence estimation. As numerical examples show, this expansion can be used on the almost entire domain of values τ2 for computing the coverage probability with error of order 10−2. In this paper, a similar asymptotic expansion is suggested, which computes the coverage probability with much smaller global error in the domain of small and moderate values of p. The accuracy of approximations is illustrated by statistical modeling data.