U-statistic based on overlapping sample spacings

TL;DR

This paper develops and analyzes a new class of U-statistics based on overlapping spacings for goodness-of-fit testing, deriving their asymptotic properties and demonstrating their efficiency and power.

Contribution

It introduces a novel U-statistic framework for overlapping spacings, extending standard theory to dependent sequences and establishing optimality and efficacy results.

Findings

The asymptotic distribution under null and alternatives is derived.

The U-statistic based on Gini's mean square difference is most powerful locally.

The test has the same efficacy as the Greenwood test.

Abstract

For testing goodness of fit, we consider a class of U-statistics of overlapping spacings of order two, and investigate their asymptotic properties. The standard U-statistic theory is not directly applicable here as the overlapping spacings form a dependent random sequence. The asymptotic distribution of the statistics under the null hypothesis and under a sequence of local alternatives are derived. In terms of the Pitman ARE, the U-statistic based on Gini's mean square difference of overlapping spacings is found to be the asymptotically locally most powerful. Interestingly, this test has the same efficacy as the Greenwood test based on overlapping spacings.