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

  • Sujitta Suraphee
  • Nongluck Viriyapong
  • Nipaporn Chutiman
  • Monchaya Chiangpradit
Keywords: Confidence sets, positive part James-Stein estimator, multivariate normal distribution, coverage probability, asymptotic expansions, third order asymptotic

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  gif.latex?\tau&space;\rightarrow&space;0 where gif.latex?\tau&space;^{2}&space;=&space; gif.latex?n is the sample size and  gif.latex?\Theta is the mean vector of the gif.latex?p-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.

Downloads

Download data is not yet available.

References

Ahmed SE, Saleh AKME, Volodin AI, Volodin IN. Asymptotic expansion of the coverage probability of James-Stein estimators. Theor Probab Appl. 2006; 51: 683-895.

Ahmed SE, Volodin AI, Volodin IN. High order approximation for the coverage probability by a confident set centered at the positive-part James-Stein estimator. Stat Probab Lett. 2009; 79: 1823-1828.

Ahmed SE, Kareev I, Suraphee S, Volodin AI, Volodin IN. Confidence sets based on the positive part James-Stein estimator with the asymptotically constant coverage probability. J Stat Comput Simulat. 2015; 85: 2506-2513.

Baranchik AJ. A family of minimax estimators of the mean of a multivariate normal distribution. Ann Math Statist.1970; 41: 642-645.

Budsaba K, Suraphee S. Addendum to “Asymptotic Expansion of the Coverage Probability of James-Stein Estimators”, Theory of Probability and Its Applications, 51(4) (2007), 683-695. J Prob Stat Sci. 2012; 10: 205-208.

Hwang JT, Casella G. Minimax confidence sets for the mean of a multivariate normal distribution. Ann Statist.1982; 10: 868-881.

Hwang JT, Casella G. Improved set estimators for a multivariate normal mean. Recent results in estimation theory and related topics. Statist Decisions suppl.1984;1: 3-16.

Stein C. Confidence sets for the mean of a multivariate normal distribution. J Roy Statist Soc, Ser B.1962; 24: 265-296.
Published
2018-07-19
Section
Articles