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U-Statistics based on spacings
Authors:David D Tung  S Rao Jammalamadaka
Institution:a ALEKS Corporation, Irvine, CA 92618, USA
b Department of Statistics and Applied Probability, University of California, Santa Barbara, CA 93106-3110, USA
Abstract:In this paper, we investigate the asymptotic theory for U-statistics based on sample spacings, i.e. the gaps between successive observations. The usual asymptotic theory for U-statistics does not apply here because spacings are dependent variables. However, under the null hypothesis, the uniform spacings can be expressed as conditionally independent Exponential random variables. We exploit this idea to derive the relevant asymptotic theory both under the null hypothesis and under a sequence of close alternatives.The generalized Gini mean difference of the sample spacings is a prime example of a U-statistic of this type. We show that such a Gini spacings test is analogous to Rao's spacings test. We find the asymptotically locally most powerful test in this class, and it has the same efficacy as the Greenwood statistic.
Keywords:Goodness-of-fit tests  Order statistics  Gini mean difference  Pitman asymptotic relative efficiency
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