Let $T_n = n^{-1} \sum^n_{i=1} c_{in}X_{in}$ be a linear combination of order statistics and put $T^\ast_n = (T_n - E(T_n))/ \sqrt{\operatorname{Var}(T_n)}$. Sufficient conditions on the $c_{in}$ and on the moments of the underlying distribution are established under which the ratio $P(T^\ast_n > x)/(1 - \Phi(x))$ tends to 1, either uniformly in the range $-A \leq x \leq c \sqrt{\ln n}(A \geq 0, c > 0)$ (moderate deviation theorem) or uniformly in the range $-A \leq x \leq o(n^\alpha)(A \geq 0)$ (Cramer type large deviation theorem). The proof relies on Helmers' approximation method and on the corresponding results for U-statistics.
Publié le : 1982-05-14
Classification:
Large deviations,
linear combinations of order statistics,
moderate deviations,
$U$-statistics,
60F10,
62G30
@article{1176993867,
author = {Vandemaele, M. and Veraverbeke, N.},
title = {Cramer Type Large Deviations for Linear Combinations of Order Statistics},
journal = {Ann. Probab.},
volume = {10},
number = {4},
year = {1982},
pages = { 423-434},
language = {en},
url = {http://dml.mathdoc.fr/item/1176993867}
}
Vandemaele, M.; Veraverbeke, N. Cramer Type Large Deviations for Linear Combinations of Order Statistics. Ann. Probab., Tome 10 (1982) no. 4, pp. 423-434. http://gdmltest.u-ga.fr/item/1176993867/