On consistent minimax distinguishability and intermediate efficiency of Cramér–von Mises test

• Published in: Journal of statistical planning and inference 2004-09, Vol.124 (2), p.453-474
• Main Authors: Inglot, Tadeusz, Ledwina, Teresa
• Format: Article
• Language: English
• Subjects: Adaptive test; Asymptotic efficiency; Consistent minimax distinguishability; Cramér–von Mises test; Decision theory; Exact sciences and technology; General topics; Intermediate efficiency; Mathematics; Probability and statistics; Sciences and techniques of general use; Statistics
• ISSN: 0378-3758; 1873-1171
• DOI: 10.1016/j.jspi.2003.03.001

Recent work on adaptive tests has raised new challenges in tests’ comparison and in asserting their optimality. Some new and unappreciated earlier approaches have been recently developed. In particular, intermediate efficiency and consistent minimax distinguishability are extensively exploited. The aim of this contribution is to provide some insight into the nature of these approaches and the relation between them. The emphasis is on the interpretation of the outcomes of both criteria. To make the presentation as simple as possible, we restrict our attention to the popular (non-adaptive) test—Cramér–von Mises test—and briefly comment only on the further use of both approaches to evaluate adaptive tests. Recent consistent minimax distinguishability considerations are, as a rule, related to some smoothness classes of alternatives. They provide an optimal rate related to the degree of smoothness. However, we prove that for Cramér–von Mises test and for the rate specific to Sobolev smoothness class, there exists a set of non-necessarily smooth alternatives providing the same rate. We are also defining some other set of alternatives guaranteeing the same rate. This shows that popular criterion of adapting to unknown smoothness is an unclear guideline hard to interpret in practice. On the other hand, we provide some evidence that intermediate efficiency and related power comparisons of Cramér–von Mises test to the best test have a nice finite sample interpretation fulfilling well established standards. Moreover, the notion of adaptive (or better—efficient) test in the intermediate setting has a similar interpretation as that postulated by Stein (Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Vol. I, 1956. pp. 187–195).