Simultaneous Estimation of Quantiles of Two Normal Populations with a Common Mean

Abstract

Let (X1;X2; : : : ;Xm) and Y = (Y1; Y2; : : : ; Yn) be independent random samples drawn from two normal populations with a common unknown mean µ and possibly unknown different variances σ_1^2 and σ_2^2 respectively. The problem of simultaneous estimation of the p^th (0 < p < 1) quantiles, θi = µ+ησ, i = 1, 2 of the two normal populations is considered with respect to a sum of quadratic loss functions. Here η=Φ^(-1) (p) and Φ(.) the cumulative distribution function of a standard normal random variable. A general result has been proved for improving the basic estimator for the quantiles. Using this result improved estimators for quantiles have been constructed. A suffcient condition for improving estimators in certain classes of affine and location equivariant estimators are obtained, as a result two complete class theorems have been proved. A massive simulation study has been carried out to compare numeri¬cally various proposed efficient estimators for the quantiles. Some practical exam¬ples have been discussed to show the applicability of our model. AMS Subject Classifcation 62C15, 62F10, 62C20