UC Santa Barbara

In this paper, we investigate the asymptotic theory for U-statistics based on higher-order sample spacings. The usual asymptotic theory for Ustatistics does not fully apply here because spacings are dependent variables. However, under the null hypothesis, the higher-order spacings can be represented by conditionally independent Gamma random variables. We exploit this idea to derive the relevant asymptotic theory both under the null hypothesis H0: F = F0 and under a sequence of close alternatives. The generalized Gini mean difference of the higher-order sample spacings is a prime example of a U-statistic of this type. It is found that the Gini mean squared difference test based on these higher-order spacings is the asymptotically locally most powerful test in this class, and has the same efficacy as the generalized Greenwood statistic based on the sum of squares of the higher-order spacings.