The Annals of Statistics

Convergence of Sample Paths of Normalized Sums of Induced Order Statistics

P. K. Bhattacharya

Full-text: Open access

Abstract

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

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

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176342823

Digital Object Identifier
doi:10.1214/aos/1176342823

Mathematical Reviews number (MathSciNet)
MR386100

Zentralblatt MATH identifier
0307.62036

JSTOR
links.jstor.org

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

Keywords
Induced order statistics Skorokhod embedding invariance principle test for regression function

Citation

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. https://projecteuclid.org/euclid.aos/1176342823


Export citation