Estimation of parameters in the tree order restriction by a randomized decision

Abstract

For estimating the smallest location parameter in the location family of distributions which are constrained by the tree ordering θ0 ≤ θi for 1 ≤ i ≤ k, the restricted maximum likelihood estimator diverges to −∞ as k→∞and therefore fails to dominate the corresponding unrestricted estimator in terms of the bias and hence the mean squared error (MSE). In this article, we propose a new procedure for the estimation of the location parameters based on a randomized decision. The proposed randomized estimator of θ0 is improved via the smooth approach to construct the better estimator which remains bounded and decreases the growth rate of its bias and MSE. We show in the case of normal distributions that the MSE of the proposed estimator of θ0 is less than that of the corresponding unrestricted estimator. By using a simulation study, the performance of the improved estimators is compared with that of the other restricted estimators in terms of three criteria (bias, MSE and coverage probability). The results show that the proposed estimator of θ0 is substantially better than that of the alternative estimators. Unlike the other procedures, the proposed method for estimating θi, i = 1, . . . , k performs well.