Breakdown Robustness of Tests

Abstract

Power and level breakdown functions of a test statistic are introduced to obtain a unified analysis of local stability (as typified by the influence function) and global stability (as illustrated by the breakdown point). The power breakdown function gives the amount of contamination of each alternative distribution that can carry the test statistic to a null value. The level breakdown function gives the amount of contamination of a null distribution that can carry the test statistic to each value in the alternative space. In many cases, the breakdown function bears an inverse relationship to the maximum bias as a function of the contamination. As the maximum bias is not invariant to one-to-one transformations, however, equivalent test statistics can have different scales for bias. A distinct advantage of the inverse scale is the invariance of the breakdown functions to such transformations. The resulting stability curves depend on the parameters in a way analogous to the power curve. The slopes of the breakdown functions for local alternatives recover the “shrinking neighborhoods” analysis, and provide an alternative derivation of the Rousseeuw-Ronchetti gross-error sensitivity for test statistics. Applications to location-scale models and oneway designs are considered. For testing location, the breakdown functions of the sign test uniformly dominate those of the Wilcoxon test, and the sign test is the uniformly most robust M test. In the one-way design, the Rao-type M-score tests have better global stability than the Wald-type M-estimate tests for contamination of null data. For local alternatives in the oneway layout, the power breakdown function of an M test factors into two terms, one depending only on the choice of score function, the other depending only on the configuration of group means.

Key Words:

• Bias function
• Breakdown function
• F test
• Influence function
• M test
• One-way design
• Sign test
• Wilcoxon test