Tests for Stochastic Orders and Mean Order Statistics

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.