On the Asymptotic Distribution of $k$-Spacings with Applications to Goodness-of-Fit Tests

Abstract

Let $X_1, \cdots, X_n$ be an ordered sample from a distribution $A_n$ on [0, 1]. The $k$-spacings $D_1(N, k), \cdots, D_N(N, k)$ are defined and the weak convergence of their empirical distribution function under a sequence of alternatives $A_n$ approaching the uniform distribution is established. This is then applied to find the limiting distribution of $W_n(g, k) = N^{-\frac{1}{2}}\Sigma^N_{i=l}(g(NkD_i(N, k)) - a)$ where $g$ is a smooth function and $k$ is fixed. The statistics $W_n(g, k)$ can be used to test the hypothesis that the observations are uniformly distributed in [0, 1]. The asymptotic relative efficiency of $W_n(g, k)$ with respect to $W_n(g, 1)$ is shown to increase without limit for several functions $g$. The test with $g(x) = x^2$ is shown to be asymptotically optimal within the class $W_n(g, k)$ for any fixed $k$. The paper extends results of Rao and Sethuraman.