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 ${\theta }_0\le {\theta }_i$ for $1\le i\le k$, the restricted maximum likelihood estimator diverges to $-\mathrm{\infty }$ as $k\to \mathrm{\infty }$ 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 ${\theta }_0$ 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 ${\theta }_0$ 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 ${\theta }_0$ is substantially better than that of the alternative estimators. Unlike the other procedures, the proposed method for estimating ${\theta }_i,i=1,\dots ,k$ performs well.

Keywords:

• Bias
• isotonic regression estimator (IRE)
• mean squared error (MSE)
• tree order restriction
• unrestricted maximum likelihood estimator (UMLE)

Acknowledgements

We would like to thank the reviewer for carefully reading this article and giving several exact and constructive comments and valuable suggestions for the better presentation of the article.

Disclosure statement

No potential conflict of interest was reported by the authors.