MULTIPLE HYPOTHESES TESTING WITH WEIGHTS

In this paper we offer a multiplicity of approaches and procedures for multiple testing problems with weights. Some rationale for incorporat ing weights in multiple hypotheses testing are discussed. Various type -I error-rates and different possible formulations are considered, for both the intersection hypothesis testing and the multiple hypotheses testing problems. An optimal per family weighted error-rate controllin g procedure a la Spjotvoll (1972) is obtained. This model serves as a vehicle for demonstrating the different implications of the approaches to weighting. Alternative approaches to that of Holm (1979) for famil y-wise error-rate control with weights are discussed, one involving an alternative procedure for family-wise error-rate control, and the oth er involving the control of a weighted family-wise error-rate. Extensi ons and modifications of the procedures based on Simes (1986) are give n. These include a test of the overall intersection hypothesis with ge neral weights, and weighted sequentially rejective procedures for test ing the individual hypotheses. The false discovery rate controlling ap proach and procedure of Benjamini & Hochberg (1995) are extended to al low for different weights.