Estimating positive powers of the scale parameters under order restriction for two normal populations with a common mean

ABSTRACT

The problem of estimating positive powers of scale parameters ${\sigma }_i^{2c},i=1,2,$ has been considered under order restriction when samples are available from two normal populations with a common mean μ. Maximum-likelihood estimators and several plug-in type estimators using some popular estimators of the common mean have been proposed. Sufficient conditions for improving equivariant estimators for the positive powers of the scale parameters under order restriction have been derived. Consequently, several improved estimators have been proposed. A numerical comparison among all the proposed estimators in the special cases (c = 1 and $c=1/2$) has been made in terms of risk values using the quadratic loss function, and recommendations are given for the use of the estimators. Finally, a real-life example has been considered to show the potential application of the model problem.

KEYWORDS:

• Equivariant estimators
• inadmissibility
• ordered parameters
• powers of scale parameters
• restricted maximum-likelihood estimators

2010 AMS SUBJECT CLASSIFICATIONS:

• 62C15
• 62F10
• 62F30

Acknowledgments

The authors would like to sincerely thank the two anonymous reviewers, the associate editor and the editor-in-chief, whose constructive and thoughtful comments on the earlier version of the manuscript led to greater improvements in the manuscript's content.

Disclosure statement

No potential conflict of interest was reported by the author(s).