Generalized Spacing-Statistics and a New Family of Non-Parametric Tests
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Random divisions of an interval arise in various context, including statistics, physics, and geometric analysis. For testing the uniformity of a random partition of the unit interval [0,1] into k disjoint subintervals of size (S_k[1],…,S_k[k]), Greenwood (1946) suggested using the squared ℓ_2-norm of this size vector as a test statistic, prompting a number of subsequent studies. Despite much progress on understanding its power and asymptotic properties, attempts to find its exact distribution have succeeded so far for only small values of k. Here, we develop an efficient method to compute the distribution of the Greenwood statistic and more general spacing-statistics for an arbitrary value of k. Specifically, we consider random divisions of {1,2,…,n} into k subsets of consecutive integers and study S_n,k^p_p,w, the pth power of the weighted ℓ_p-norm of the subset size vector S_n,k=(S_n,k[1],…,S_n,k[k]) for arbitrary weights w=(w_1,…,w_k). We present an exact and quickly computable formula for its moments, as well as a simple algorithm to accurately reconstruct a probability distribution using the moment sequence. We also study various scaling limits, one of which corresponds to the Greenwood statistic in the case of p=2 and w=(1,…,1), and this connection allows us to obtain information about regularity, monotonicity and local behavior of its distribution. Lastly, we devise a new family of non-parametric tests using S_n,k^p_p,w and demonstrate that they exhibit substantially improved power for a large class of alternatives, compared to existing popular methods such as the Kolmogorov-Smirnov, Cramer-von Mises, and Mann-Whitney/Wilcoxon rank-sum tests.
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