## The Annals of Statistics

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

### Convergence of Sample Paths of Normalized Sums of Induced Order Statistics

#### 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