Data driven versions of pearson's chisquare test for uniformity

Abstract

The test statistic of Pearson's Chi-square test for uniformity can be seen as the L ₂-distance between the null density and the histogram density estimator. The power of this test depends heavily on the number of histogram cells. Recommended procedures for choosing this number usually exploit the knowledge of the class of alternatives. We present how the Schwarz's Bayesian Information Criterion and certain minimum complexity criteria can be used for selecting the number of classes for Pearson's Chi-square test. These criteria allow us to make a choice depending only on the observed data. We compare the powers of the resulting data driven tests with the power of Chi-square test by means of Monte Carlo simulations. We investigate also a test based on the L ₂-distance between the null density and the mixture of histogram density estimators, introduced by J. Rissanen. This test turns out to be much better than Pearson's test, and it is competitive and comparable to other known test procedures.

Keywords:

• Goodness-of-fit
• Pearson's-square test
• Schwarz's BIC criterion
• Stochastic complexity
• MDL criterion
• Monte Carlo simulations