Tests of Symmetry in Three-dimensional Contingency Tables Based on Phi-divergence Statistics

In this paper we introduce a family of test statistics for testing complete symmetry in three-dimensional contingency tables based on phi- divergence families. These test statistics yield the likelihood ratio test and the Pearson test statistics as special cases. Asymptotic distribution for the new test statistics are derived under both the null and the alternative hypotheses. A simulation study is presented to show that some new statistics offer an attractive alternative to the classical Pearson and likelihood ratio test statistics for this problem of complete symmetry.     

Spatial Clustering Tests Based on the Domination Number of a New Random Digraph Family

We use the domination number of a parametrized random digraph family called proportional-edge proximity catch digraphs (PCDs) for testing multivariate spatial point patterns. This digraph family is based on relative positions of data points from various classes. We extend the results on the distribution of the domination number of proportional-edge PCDs, and use the domination number as a statistic for testing segregation and association against complete spatial randomness. We demonstrate that the domination number of the PCD has binomial distribution when size of one class is fixed while the size of the other (whose points constitute the vertices of the digraph) tends to infinity and has asymptotic normality when sizes of both classes tend to infinity. We evaluate the finite sample performance of the test by Monte Carlo simulations and prove the consistency of the test under the alternatives. We find the optimal parameters for testing each of the segregation and association alternatives. Furthermore, the methodology discussed in this article is valid for data in higher dimensions also.