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 incorporating 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 controlling procedure a la Spjotvoll (1972) is obtained. This model serves as a vehicle for demonstrating the different implications of the approaches to weighting. Alternative approach es to that of Holm (1979) for family‐wise error‐rate control with weights are discussed, one involving an alternative procedure for family‐wise error‐rate control, and the other involving the control of a weighted family‐wise error‐rate. Extensions and modifications of the procedures based on Simes (1986) are given. These include a test of the overall intersec tion hypothesis with general weights, and weighted sequentially rejective procedures for testing the individual hypotheses. The false discovery rate controlling approach and procedure of Benjamini & Hochberg (1995) are extended to allow for different weights.