Bonferroni and Holm Approximations for Šidák and Holland–Copenhaver q-Values

Holm ’ s Sequential Bonferroni Procedure

The more statistical tests we perform, the more likely we are to reject the null hypothesis when it is true (i.e., a “false alarm,” also called a “Type 1” error). This is a consequence of the logic of hypothesis testing: We reject the null hypothesis for rare events, and the larger the number of tests, the easier it is to find rare events which are false alarms. This problem is called the infla...

A note on adaptive Bonferroni and Holm procedures under dependence

Hochberg & Benjamini (1990) first presented adaptive procedures for controlling familywise error rate. However, until now, it has not been proved that these procedures control the familywise error rate. We introduce a simplified version of Hochberg & Benjamini’s adaptive Bonferroni and Holm procedures. Assuming a conditional dependence model, we prove that the former procedure controls the fami...

Support Approximations Using Bonferroni-Type Inequalities

Compatible simultaneous lower confidence bounds for the Holm procedure and other Bonferroni-based closed tests.

We consider the problem of simultaneously testing multiple one-sided null hypotheses. Single-step procedures, such as the Bonferroni test, are characterized by the fact that the rejection or non-rejection of a null hypothesis does not take the decision for any other hypothesis into account. For stepwise test procedures, such as the Holm procedure, the rejection or non-rejection of a null hypoth...

Optimized Bonferroni Approximations of Distributionally Robust Joint Chance Constraints

A distributionally robust joint chance constraint involves a set of uncertain linear inequalities which can be violated up to a given probability threshold , over a given family of probability distributions of the uncertain parameters. A conservative approximation of a joint chance constraint, often referred to as a Bonferroni approximation, uses the union bound to approximate the joint chance ...