The Annals of Statistics

On the Asymptotic Distribution of Quadratic Forms in Uniform Order Statistics

Peter Guttorp and Richard A. Lockhart

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The asymptotic distribution of quadratic forms in uniform order statistics is studied under contiguous alternatives. Using minimal conditions on the sequence of forms, the limiting distribution is shown to be the convolution of a sum of weighted noncentral chi-squares and a normal variate. The results give approximate distribution theory even when no limit exists. As an example, high-order spacings statistics are shown to have trivial asymptotic power unless the order of the spacings grows linearly with the sample size. The results are derived from a modification of an invariance principle for quadratic forms due to Rotar, which we prove by martingale central limit methods.

Article information

Ann. Statist., Volume 16, Number 1 (1988), 433-449.

First available in Project Euclid: 12 April 2007

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Zentralblatt MATH identifier


Primary: 62E20: Asymptotic distribution theory
Secondary: 62F05: Asymptotic properties of tests

Goodness of fit martingale central limit theorem asymptotic relative efficiency


Guttorp, Peter; Lockhart, Richard A. On the Asymptotic Distribution of Quadratic Forms in Uniform Order Statistics. Ann. Statist. 16 (1988), no. 1, 433--449. doi:10.1214/aos/1176350713.

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