Statistical Test for Bivariate Uniformity

The purpose of the multidimension uniformity test is to check whether the underlying probability distribution of a multidimensional population differs from the multidimensional uniform distribution. The multidimensional uniformity test has applications in various fields such as biology, astronomy, and computer science. Such a test, however, has received less attention in the literature compared with the univariate case. A new test statistic for checking multidimensional uniformity is proposed in this paper. Some important properties of the proposed test statistic are discussed. As a special case, the bivariate statistic test is discussed in detail in this paper. The Monte Carlo simulation is used to compare the power of the newly proposed test with the distance-to-boundary test, which is a recently published statistical test for multidimensional uniformity. It has been shown that the test proposed in this paper is more powerful than the distance-to-boundary test in some cases. 1. Introduction Testing uniformity in the univariate case has been studied by many researchers, whereas the multidimensional uniformity test seems to have received less attention in the literature. Testing whether a pattern of points in the multidimensional space is distributed uniformly has applications in many fields such as biology, astronomy, and computer science. A commonly used goodness-of-fit test for uniformity is the chi-square test [1]. Theoretically, the chi-square test can be applied for any multivariate distribution test. However, the problem for the chi-square test is the arbitrariness of cell limits determination. Another problem for the chi-square test is that the power of the chi-square test is usually low. Some other well-known methods for univariate goodness-of-fit tests are the Kolmogorov-Smirnov test [2, 3], Anderson-Darling test [4], and the Cramer-von Mises test [5]. Justel et al. [6] proposed a multivariate goodness-of-fit test based on the idea of the Kolmogorov-Smirnov test. By using the Rosenblatt’s transformation, they reduced the multivariate case to univariate case. The test statistic they used has distribution free property and can be applied to any dimensional case. The problem for that method is that the computation of test statistic is complicated especially for over two dimensions. Liang et al. [7] proposed several statistical tests for testing uniformity in multivariate case. Those tests used the number-theoretic and quasi-Monte Carlo method for measuring the discrepancy of the points in multidimensional unit. Berrendero et al. [8] proposed a