The Problem of Simultaneous Inference

Abstract

We introduce the problem of simultaneous statistical inference, with particular emphasis on testing multiple hypotheses. After a historic overview, general notation for the whole work is set up and different sources of multiplicity are distinguished. We define a variety of classical and modern type I and type II error rates in multiple hypotheses testing, analyze some relationships between them, and consider different ways to cope with structured systems of hypotheses. Relationships between multiple testing and other simultaneous statistical inference problems, in particular the construction of confidence regions for multi-dimensional parameters, as well as selection, ranking and partitioning problems, are elucidated. Finally, a general outline of the remainder of the work is given.