A comparison of tests for randomness

摘要

This paper compares five distance-based statistics (Hopkins, Cox-Lewis, Eberhardt and two T-square statistics) which reflect the degree to which a set of d-dimensional points is random. The objectives are to identify the most promising statistic for intensive study and to establish performance bounds for situations involving little prior information. The sizes and powers of tests of several aggregation models vs. randomness and several regularity models vs. randomness are estimated using theoretical thresholds to define tests of hypothesis. Both hypercubic and hyperspherical sampling windows are examined with a sampling frame of uniform thickness. The Hopkins statistic is the best overall statistic, but the thresholds must be established by Monte Carlo means for more than five dimensions. The sampling window has a greater effect on tests for regularity than on tests for aggregation.