A Bootstrap Revival of Some Nonparametric Distance Tests

Abstract

Several tests based on the empirical measure have been proposed to test independence of variables, goodness of fit, equality of distributions, rotational invariance, and so forth. These tests have excellent power properties, but critical values are difficult, if not impossible, to obtain. Furthermore, these tests usually assume that the data are real-valued with continuous distributions. Here, critical values are determined by bootstrapping and the resulting tests are shown to have the correct asymptotic level under minimal assumptions. For example, given data X_i = (X _i,1, …, X_i,d ), i = 1, …, n, it may be desired to test independence of the d components. The proposed test compares the empirical measure and the product of its marginals by taking a supremum over an appropriate Vapnik-Cervonenkis class of sets. No assumptions are made on the probability distribution of the data or on the space in which it lives; indeed, some components may be discrete, some continuous, and others categorical. Similar results are obtained for other examples. Consistency of the tests is obtained against all alternatives. A modest simulation study shows that the bootstrap works satisfactorily for moderate sample sizes.

Key Words:

• Distance tests
• Goodness of fit
• Nonparametric tests
• Testing equality of distributions
• Testing for rotational invariance
• Testing independence
• Vapnik-Cervonenkis classes