An optimal statistic based on higher order gaps

The family of statistics based on mth-order gaps from a uniform sample, obtained by summing a suitably regular function of each gap, is investigated. Holst (1979) has established asymptotic normality together with explicit expressions for the mean and variance. This is extended to samples from distributions whose perturbations from uniformity HhrinV as sample size grows. From these results, Pitman asymptotic relative efficiencies can be calculated, and it is shown that the sum of squares of gaps is an optimal statistic. The properties of this statistic are presented, together with a comparison to the already well treated sum of log gaps. It is shown that the two become indistinguishable as m, the order of the gaps, grows.

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