Some count-based nonparametric tests for circular symmetry of a bivariate distribution

ABSTRACT

In this paper, we revisit the problem of testing of the hypothesis of circular symmetry of a bivariate distribution. We propose some nonparametric tests based on sector counts. These include tests based on chi-square goodness-of-fit test, the classical likelihood ratio, mean deviation, and the range. The proposed tests are easy to implement and the exact null distributions for small sample sizes of the test statistics are obtained. Two examples with small and large data sets are given to illustrate the application of the tests proposed. For small and moderate sample sizes, the performances of the proposed tests are evaluated using empirical powers (empirical sizes are also reported). Also, we evaluate the performance of these count-based tests with adaptations of several well-known tests such as the Kolmogorov–Smirnov-type tests, tests based on kernel density estimator, and the Wilcoxon-type tests. It is observed that among the count-based tests the likelihood ratio test performs better.

KEYWORDS:

• Circular symmetry
• Count statistics
• Empirical power
• Nonparametric test

MATHEMATICS SUBJECT CLASSIFICATION:

• 62G10
• 62G20
• 62H15

Acknowledgment

The authors are grateful to reviewers and the editor for their constructive suggestions that have helped to improve the earlier version of the manuscript. The second author wishes to thank the University of Botswana for granting him the research leave during which a part of the research work was undertaken.