We propose general single-step and step-down multiple testing procedures for controlling Type I error rates defined as arbitrary parameters of the distribution of the number of Type I errors (e.g., generalized family-wise error rate). The method is based on a null distribution for the test statistics, rather than a data generating null distribution, and provides asymptotic control of the Type I error rate without the need for conditions such as subset pivotality.
For general null hypotheses, corresponding to submodels for the data generating distribution, the proposed null distribution is the asymptotic distribution of the vector of null-value shifted and scaled test statistics. In the special case of family-wise error rate control, this general approach reduces to the minP and maxT procedures based on minima of p-values and maxima of test statistics, respectively. Resampling procedures (e.g., based on the non-parametric or model-based bootstrap) are provided to conveniently obtain consistent estimators of the null distribution.
Applications to the identification of differentially expressed genes in microarray experiments are presented.

Joint work with Mark J. van der Laan and Katherine S. Pollard.