Higher-order expansions and efficiencies of tests based on spacings

Statistics based on spacings, or the gaps between points, have been widely used in many contexts, primarily in testing goodness of fit. This paper derives Edgeworth-type asymptotic expansions for the sum of functions of s-step spacings where s, the order of spacings, may increase together with the sample size n. When s is fixed, it is known that only the Greenwood test, based on the sum of squares of these spacings, is first-order asymptotically efficient. In contrast, it is shown here that if s goes to infinity, there exist many other tests which are first-order efficient. We introduce and study the second-order efficiency of such tests and show that if s is sufficiently large relative to n, the Greenwood test is no longer second-order efficient. Interestingly, we see that the common phenomenon of first-order efficiency implying second-order efficiency does not hold true in this situation.

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