On an estimator of normal population mean and UMVU estimation of its relative efficiency

Abstract

For estimating the mean μ of a normal population with variance σ 2, Searles [J. Am. Stat. Assoc. 59 (1964) 1225] provided the minimum mean squared error (MMSE) estimator (1+σ 2/(nμ 2)) −1 x ̄ in the class of estimators of the type: k x ̄ . However, as σ/ μ is seldom known, this MMSE estimator is not very useful, in practice. Srivastava [Indian Soc. Agric. Stat. 26 (1974) 33 and Metrika 27 (1980) 99], therefore, proposed the correspondingly computable estimator t= x ̄ /(1+s 2/(n x ̄ 2)) , and showed that it is more efficient than the usual estimator x̄ whenever σ 2/( nμ 2) is at least 0.5. Nevertheless, the relevant gain in efficiency would be still unknown in as much as it invlolves the unknown population parameters μ and σ 2. Srivastava and Singh [Stat. Probab. Lett. 10 (1990) 241] provided an uniformly minimum variance unbiased estimate of the relative efficiency ratio: E( x ̄ −μ) 2/E(t−μ) 2 to help determine the usefulness of the estimator t over the usual sample mean estimator x̄, in practice. Very often the coefficient of variation of the usual sample mean estimator x̄ (more stable than the original study variable X), and hence its sample counterpart (estimated value) could be rather low, say less than 0.5, so that s 2/n x ̄ 2(=v, say) is less than 1/4. For such situations, the present paper proposes t ∗= x ̄ /(1−s 2/(n x ̄ 2)) , and studies it on the lines similar to those t has been studied on in [Stat. Probab. Lett. 10 (1990) 241].