3 Robust inference: The approach based on influence functions

Abstract

This chapter discusses the concept of robust inference using an approach known as influence functions. It also describes the two key tools, the influence function and the breakdown point, and their application to the inference problem. They can be used to investigate the local stability and the global reliability of a test or confidence interval. Moreover, they provide the basis for constructing new robust tests and confidence intervals. The chapter presents robust tests for general parametric models. Robust analogues of likelihood ratio, Wald, and scores (Rao) tests are derived in the chapter. Robust inference procedures for linear and nonlinear models are discussed in the chapter. Some numerical results show the finite sample performance of these robust procedures. The purpose in robust testing is twofold. First, the level of a test should be stable under small, arbitrary departures from the null hypothesis (robustness of validity). Secondly, the test should still have a good power under small arbitrary departures from specified alternatives (robustness of efficiency). For confidence intervals, these criteria translate to coverage probability and length of the interval.