Pyramidal Distribution: An Alternative Distribution for Bivariate Uniformity Test Power Comparison

Abstract

The purpose of the multi-dimensional uniformity test is to check whether the underlying probability distribution, from which a random sample is drawn, differs from the multi-dimensional uniform distribution. The multi-dimensional uniformity test has applications in various fields such as biology, astronomy and computer science. Various statistical tests have been proposed for the multivariate uniformity. As a special case, the bivariate statistic test is discussed in this paper. To evaluate the performance of the uniformity tests, power comparison is the technique for selecting appropriate statistical test. The Monte Carlo simulations usually used to compare the power of the tests. Berrendero, Cuevas and Vazquez-Grande proposed a distance-to-boundary test, which was one of the latest published statistical tests for multi-dimensional uniformity [J. R. Berrendero, A. Cuevas and F. Vazquez-Grande, Testing multivariate uniformity: The distance-to-boundary method, Canad. J. Stat. 34 (2006) 693–707]. Chen and Hu proposed another test for multivariate uniformity [Z. Chen and T. Hu, Statistical test for bivariate uniformity, Adv. Stat. 2014 (2014) 740831]. Power comparison was conducted to compare these tests. In order to get more convincing power comparison results, more alternative distributions should be used when power study is conducted. This paper proposes a new bivariate distribution, named the pyramidal distribution, with support set [Formula: see text]. This distribution is quite flexible so that it can be used to produce different shapes of bivariate distributions. Because of that, the proposed distribution can be used as an alternative distribution in power comparison for bivariate uniformity test.