The Annals of Statistics

Convergence of Sample Paths of Normalized Sums of Induced Order Statistics

P. K. Bhattacharya

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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.

Article information

Ann. Statist., Volume 2, Number 5 (1974), 1034-1039.

First available in Project Euclid: 12 April 2007

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Zentralblatt MATH identifier


Primary: 60F99: None of the above, but in this section
Secondary: 62E20: Asymptotic distribution theory

Induced order statistics Skorokhod embedding invariance principle test for regression function


Bhattacharya, P. K. Convergence of Sample Paths of Normalized Sums of Induced Order Statistics. Ann. Statist. 2 (1974), no. 5, 1034--1039. doi:10.1214/aos/1176342823.

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