Significance levels for multiple tests

✍ Scribed by Sergiu Hart; Benjamin Weiss

• Book ID: 104302589
• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 305 KB
• Volume: 35
• Category: Article
• ISSN: 0167-7152
• DOI: 10.1016/s0167-7152(96)00214-3

✦ Synopsis

Let X 1 ..... X, be n random variables, with cumulative distribution functions F1, ..., F,. Define ¢i := Fi(Xi) for all i, and let ~(1) ~< ... ~< ~(,) be the order statistics of the (~)~. Let cq ~< ... ~< ~. be n numbers in the interval [0, 1]. We show that the probability of the event R := {(") ~< ct~ for all 1 ~< i ~< n} is at most min~n~Ji }. Moreover, this bound is exact: for any given n marginal distributions (F~)~, there exists a joint distribution with these marginals such that the probability of R is exactly min~nct~/i}. This result is used in analyzing the significance level of multiple hypotheses testing. In particular, it implies that the ROger tests dominate all tests with rejection regions of type R as above.