On two-stage and three-stage sampling procedures for fixed-width interval estimation of the common variance of correlated normal random variables.

摘要

Since the time of Stein's infamous, and invaluable, presentation of a two-stage sampling procedure for fixed-width interval estimation of the mean of i.i.d. normal random variables in the presence of unknown variance, two-stage and three-stage procedures for fixed-width interval estimation, with respect to normal populations, have primarily focused on estimation pertaining to means when variances are unknown. In this study, we consider such procedures for fixed-width interval estimation of the common variance of correlated normal random variables when all correlations are unknown. In particular, we consider each of the scale-equivariant estimators of the common variance of Ghezzi & Zacks (2004), for both the equicorrelated and arbitrary correlation cases. We begin the study with introduction and further exploration of Ghezzi's and Zacks' estimators, followed by the development and exploration of Stein-like two-stage procedures, incorporating the estimators into the sampling procedures in various fashions, providing relevant theory where possible and simulation results throughout. We then generalize our two-stage procedures to two different types of three-stage procedures, one of our own devise, and the other an adaptation of the modified two-stage procedure of Ghosh & Mukhopadhyay (1981), similarly investigating the incorporation of Ghezzi's and Zacks' estimators, again providing simulation results for every procedure. We conclude the study with final comparisons of the best-performing procedures, along with a brief summary of our findings and suggestions for further research.