When testing that a sample of n points in the unit hypercube [0, 1]d comes from a uniform distribution, the Kolmogorov-Smirnov and the Cramér-von Mises statistics are simple and well-known procedures. To encompass these measures of uniformity, Hickernell (1996, 1998) introduced the so-called generalized Lp−discrepancies. These discrepancies can be used in numerical integration by Monte Carlo and quasi-Monte Carlo methods, design of experiments, uniformity testing and goodness of fit tests. The aim of this paper is to derive the strong and weak asymptotic properties of these statistics.
Statistical Properties of Generalized Discrepancies and Related Quantities
SERI, RAFFAELLO
2004-01-01
Abstract
When testing that a sample of n points in the unit hypercube [0, 1]d comes from a uniform distribution, the Kolmogorov-Smirnov and the Cramér-von Mises statistics are simple and well-known procedures. To encompass these measures of uniformity, Hickernell (1996, 1998) introduced the so-called generalized Lp−discrepancies. These discrepancies can be used in numerical integration by Monte Carlo and quasi-Monte Carlo methods, design of experiments, uniformity testing and goodness of fit tests. The aim of this paper is to derive the strong and weak asymptotic properties of these statistics.
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