The kolmogorov-smirnov statistics in variation testing

Abstract

Traditionally, the Kolmogorov-Smirnov statistics are used as general goodness-of-fit tests which are known to be more sensitive to location than to scale alternatives. In this paper a statistic is defined in terms of the one-sample Kolmogorov statistics, and a distribution free scale test for the one-sample problem is developed. The understanding of this new scale statistic aids in the interpretation of p-p plots with respect to differences in scale and location alternatives. The critical values of the proposed test statistic and its limiting distribution are given. Power studies based on computer simulation indicate that the proposed test statistic has, for scale alternatives, power which is never less and usually greater than the traditional Kolmogorov goodness-of-fit test. The Bahadur asymptotic relative efficiency is shown to be the same as a nonparametric scale test proposed by Gelzer and Pyke (1965).

Keywords:

• Kolmogorov-smirnov statistics
• variance tests; quantile-quantile plots
• one-sample problem
• Bahadur asymptotic efficiency