Small-sample study of the use of mid-p power divergence goodness-of-fit tests

Abstract

Asymptotic goodness-of-fit tests based on large sample theory have been shown to be unreliable in small sample designs and sparse data structures. In these situations, it is widely believed that exact version tests are dependable. However, exact goodness-of-fit tests that are based on discrete probabilities are conservative and the calculation of the exact distribution may be computationally intractable. In this article, we present a method, which is based on recursive polynomial multiplication, for performing exact analysis of goodness-of-fit tests. This method has been shown to be more efficient and accurate than an existing fast Fourier transform method. To overcome the conservativeness problem, we suggest the use of mid–P version goodness–of–fit tests. The small sample properties of the asymptotic, the exact, and the mid–P version goodness–of–fit tests are examined via the recursive polynomial multiplication method. Four commonly used power divergence statistics and six null hypotheses ranging from the equiprobable to the extremely skew are considered. The actual significance level behaviors under various null hypotheses suggest that as an anti–conservative test, the mid–P version tests behave fairly well. Nevertheless, it is not the case for the asymptotic and the exact tests. Among the four statistics considered, the exact power comparisons under a specific alternative to various null hypotheses show that the power divergence statistic with λ= 2/3 and the Pearson’s chi–squared statistic (λ = 1) are generally the most preferable.

Keywords:

• Actual significance level
• asymptotic test
• exact power
• exact test
• mid-P test
• power divergence statistic