James–Stein confidence sets: equal area approach in the global approximation for the coverage probability

In paper [Ahmed S. E., Saleh A. K. MD. E., Volodin A. I., Volodin I. N. Asymptotic expansion of the coverage probability of James–Stein estimators // Theory Probab. Appl. – 2007. – V. 51. – P. 683–695] an asymptotic expansion of the coverage probabilities for the James–Stein confidence sets was constructed, which is accurate both for large and small values of the noncentrality parameter $\tau^2$ – the sum of the squares of the means of $p\geq3$ normal distributions. As numerical illustrations show, the expansion might be used almost in the entire area of the values of $\tau^2$ with the error of the order $10^{-2}$. In the present article a similar asymptotic expansion is suggested, whose global error is significantly less in the area of small and moderate values of $p$. The accuracy of the obtained results is shown by the Monte-Carlo statistical simulations.