Locally Most Powerful Rank Tests for Comparison of Two Failure Rates Based on Multiple Type-II Censored Data

Abstract

This article deals with the locally most powerful rank tests for testing the hypothesis that two failure rates are equal against the alternative that one failure rate is greater than the other, when the combined ordered sample is multiple Type-II censored. A modified version of the Dupač and Hájek (1969) theorem is used to establish their asymptotic normality under fixed alternative since the scores generating functions associated with these rank test statistics have a finite number of jump discontinuities. The modified version that leads to a simpler centering constant, is proved by Dupač (1970) using the results of Hájek (1968). The Pitman AREs of these rank tests based on censored data relative to the corresponding tests based on complete data are obtained under some Lehmann-type alternative distributions such that their failure rates dominate the failure rates of the respective null distributions. The AREs are computed numerically for single (left or right) and double censored data, and the extent of loss due to these censoring schemes is discussed. The rank tests considered here include among them the Mann-Whiney-Wilcoxon (MWW) test, the Savage test, and the linear combination of these two tests. In the case of all the tests, except the MWW test, it is found that the loss of efficiency due to left censoring is considerably less than that due to right censoring. In the case of finite samples, Monte Carlo simulation results showing the empirical levels and empirical powers against some Lehmann alternatives are presented.

Keywords:

• Asymptotic relative efficiency
• Failure rates
• Lehmann-type alternatives
• Loss of efficiency
• Multiple Type-II censoring
• Rank tests

Mathematics Subject Classification:

• 62G10
• 62N01

Acknowledgments

The authors are thankful to the Associate Editor for bringing to their attention two important references and for providing some useful suggestions. They are also grateful to the referee for his many valuable comments and suggestions which led to a substantial improvement in the presentation of this article.