Estimating Common Mean of a Bivariate Normal Population with Order Restricted Variances

Abstract

Let (X₁, Y₁), (X₂, Y₂),…,(X _n , Y _n ) be a random sample of size n from a bivariate normal population with a common mean μ variances ${\sigma }_1^2$ and ${\sigma }_2^2$ and correlation coefficient ρ The estimation of the common mean μ has been considered when it is known a priori that the variances are ordered, that is, ${\sigma }_1^2\le {\sigma }_2^2$ A new estimator has been proposed which dominates the maximum likelihood estimator (without restrictions) stochastically and also in terms of Pitman measure of closeness. Further an inadmissibility condition has been obtained for affine equivariant estimators when the variances are ordered. A numerical comparison of various estimators has been carried out.