Tests for Stochastic Orders and Mean Order Statistics

Abstract

The aim of this article is to emphasize the fact, not observed previously in the literature, that many discrepancy measures used in tests related to different stochastic orders can be expressed as expectations of order statistics. In this way, we provide a new meaning to the corresponding test statistics which allows us to understand better, and potentially improve, the testing procedures. As illustration, we consider tests to detect overdispersion with respect to a specific probability model. In this setting, a test for the Weibull distribution is discussed in detail.

Keywords:

• Convex order
• Dilation order
• Dispersive order
• DMRL
• Exponential distribution
• HNBUE
• Hypothesis testing
• NBUE
• Order statistics
• Total time on test transform order
• Weibull distribution

Mathematics Subject Classification:

• Primary 60E15
• 60E99
• 62E99
• 62G10
• 62E20
• Secondary 62N05

Acknowledgments

The authors would like to thank the Editor and the reviewers for their comments which help to improve the first version of the article.

This research was supported by the Spanish MCyT, grants MTM2007-66632, MTM2010-17366, and MTM2011-27248, and Comunidad de Madrid-UAM grant CCG10-UAM/ESP-5494.