A note on the pitman estimator of ordered normal means when the variances are unequal

Independent random samples are drawn from k(>= 2) normal populations having means theta(1), . . ., theta(k); theta(1) <= theta(2) <= . . . <= theta(k) and known variances sigma(2)(1) , . . . , sigma(2)(k). Estimation of theta = (theta(1) ,..., theta(k)) with respect to the norm squared error loss is considered. Let delta(p) be the analog of the Pitman estimator of theta, that is, the generalized Bayes estimator of theta with respect to the uniform prior on the restricted space Omega = {theta : theta(1) <= theta(2) <= (...) <= theta(k)}. It has been proved earlier that when the sigma(2)(i)'s are equal, delta(p) is minimax. Here we exhibit for k = 3, that when sigma(2)(i)'s are unequal, the minimaxity of delta(p) may fail to hold. Furthermore, the risk performance of delta(p) is compared numerically with that of the restricted maximum likelihood estimator delta(MLE) and the usual estimator X = (X-1, X-2, X-3).