Joseph P. Romano

Stanford University

**Abstract**

Consider the multiple testing problem of testing $k$ null

hypotheses, where the unknown family of distributions is

assumed to satisfy a certain monotonicity assumption.

Attention is restricted to procedures that control the

familywise error rate (FWE) in the strong sense and which

satisfy a monotonicity condition. Under these assumptions,

we prove certain maximin optimality results for the

well-known step-down and step-up procedures.In the second part, we consider the general problem of

contructing methods that control the FWE. In order to

improve upon the Bonferroni method or Holm's (1979) step-down

method, Westfall and Young (1993) make effective use of

resampling to construct step-down methods that implicitly

estimate the dependence structure of the test statistics.

However, their methods depend on an assumption called subset

pivotality.We will show how to construct methods that control the FWE,

both in finite and large samples. A key ingredient is

monotonicity of critical values which allows one to effectively

reduce the multiple testing problem of controlling the FWE to

the single testing problem of controlling the probability of

a Type 1 error.(The first part is joint work with Erich Lehmann and Juliet

Shaffer, and the second part is joint work with Michael Wolf.)