Uniformly minimum variance unbiased estimator of efficiency ratio in estimation of normal population mean

Abstract

For the estimation of the mean μ of a normal population with unknown variance σ 2, Searles (1964) provides the minimum mean squared (MMSE) estimator (1 + σ 2/( nμ 2)) −1 x in the class of all estimators of the type x . This MMSE estimator however is not computable in practice if σ/μ is unknown. Srivastava (1980) showed that the corresponding computable estimator t = (1 + s 2/( n x 2)) x is more efficient than the usual estimator x whenever σ 2/( nμ 2) is at least 0.5. However, the gain in efficiency is a function of μ and σ 2, and therefore remains unknown. This note provides a uniformly minimum variance unbiased estimate of the exact efficiency ratio E( t − μ) 2/ E( x − μ) 2 to help determine the usefulness of t over x in practice.