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Activity Number:
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245
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Type:
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Contributed
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Date/Time:
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Tuesday, August 5, 2003 : 10:30 AM to 12:20 PM
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Sponsor:
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Section on Nonparametric Statistics
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| Abstract - #301293 |
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Title:
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Asymptotic Properties of Empirical Quadratic Distances
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Author(s):
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Ke Yang*+ and Shu-Chuan Chen and Surajit Ray and Bruce G. Lindsay and Marianthi Markatou
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Companies:
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Pennsylvania State University and Pennsylvania State University and Pennsylvania State University and Pennsylvania State University and Columbia University
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Address:
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422A Thomas Bldg., University Park, PA, 16802-2112,
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Keywords:
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quadratic distance ; spectral decomposition ; distance kernel ; weighted sum of chi-squares
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Abstract:
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The Kramer-von Mises goodness-of-fit statistic is a well-known test statistic based on the distance between the empirical distribution function and the null distribution function. However, these statistics are defined only for univariate data. We will identify a large class of quadratic form empirical distance measures which can be defined on the sample spaces of arbitrary dimension. The kernels for these distances can be constructed so as to make the distance calculations simple for appropriate null hypotheses. We show that under very mild conditions on the kernel, the limiting distribution of the empirical distance equals the distribution of a sum of weighted independent \chi^2 random variables.The proof is elementary, not requiring empirical process theory but it does employ very general spectral decomposition result for the kernel of the distance. Distance kernels will be given for spherical and binary sequence data that generalize the nice properties of the normal kernel in n-dimensional Euclidean space.
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