## The Annals of Statistics

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

P. K. Bhattacharya

#### 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

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