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September, 1974 Convergence of Sample Paths of Normalized Sums of Induced Order Statistics
P. K. Bhattacharya
Ann. Statist. 2(5): 1034-1039 (September, 1974). DOI: 10.1214/aos/1176342823


The main result in this paper concerns the limiting behavior of normalized cumulative sums of induced order statistics obtained from $n$ independent two-dimensional random vectors, as $n$ increases indefinitely. By means of a Skorokhod-type embedding of these cumulative sums on Brownian Motion paths, it is shown that under certain conditions the sample paths of these normalized sums converge in a certain sense to a process obtained from the Brownian Motion by a transformation of the time-axis. This yields an invariance principle similar to Donsker's. In particular, the asymptotic distribution of the supremum of the absolute values of these normalized cumulative sums is obtained from a well-known result for the Brownian Motion. Large sample tests of a specifieds regression function are obtained from these results.


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P. K. Bhattacharya. "Convergence of Sample Paths of Normalized Sums of Induced Order Statistics." Ann. Statist. 2 (5) 1034 - 1039, September, 1974.


Published: September, 1974
First available in Project Euclid: 12 April 2007

zbMATH: 0307.62036
MathSciNet: MR386100
Digital Object Identifier: 10.1214/aos/1176342823

Primary: 60F99
Secondary: 62E20

Keywords: induced order statistics , invariance principle , Skorokhod embedding , test for regression function

Rights: Copyright © 1974 Institute of Mathematical Statistics


Vol.2 • No. 5 • September, 1974
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