On the Asymptotic Distribution of Quadratic Forms in Uniform Order Statistics

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 resul...

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Bibliographic Details
Published in:The Annals of statistics 1988-03, Vol.16 (1), p.433-449
Main Authors: Guttorp, Peter, Lockhart, Richard A.
Format: Article
Language:English
Subjects:
62E20
62F05
asymptotic relative efficiency
Distribution functions
Distribution theory
Eigenfunctions
Eigenvalues
Exact sciences and technology
Goodness of fit
Infinity
martingale central limit theorem
Martingales
Mathematical theorems
Mathematics
Null hypothesis
Probability and statistics
Random variables
Sciences and techniques of general use
Statistical theories
Statistics
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Summary: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.
ISSN:0090-5364
2168-8966
DOI:10.1214/aos/1176350713