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.
Publié le : 1974-09-14
Classification:
Induced order statistics,
Skorokhod embedding,
invariance principle,
test for regression function,
60F99,
62E20
@article{1176342823,
author = {Bhattacharya, P. K.},
title = {Convergence of Sample Paths of Normalized Sums of Induced Order Statistics},
journal = {Ann. Statist.},
volume = {2},
number = {1},
year = {1974},
pages = { 1034-1039},
language = {en},
url = {http://dml.mathdoc.fr/item/1176342823}
}
Bhattacharya, P. K. Convergence of Sample Paths of Normalized Sums of Induced Order Statistics. Ann. Statist., Tome 2 (1974) no. 1, pp. 1034-1039. http://gdmltest.u-ga.fr/item/1176342823/