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May, 1982 Cramer Type Large Deviations for Linear Combinations of Order Statistics
M. Vandemaele, N. Veraverbeke
Ann. Probab. 10(2): 423-434 (May, 1982). DOI: 10.1214/aop/1176993867


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.


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M. Vandemaele. N. Veraverbeke. "Cramer Type Large Deviations for Linear Combinations of Order Statistics." Ann. Probab. 10 (2) 423 - 434, May, 1982.


Published: May, 1982
First available in Project Euclid: 19 April 2007

zbMATH: 0482.60026
MathSciNet: MR647514
Digital Object Identifier: 10.1214/aop/1176993867

Primary: 60F10
Secondary: 62G30

Keywords: $U$-statistics , large deviations , linear combinations of order statistics , Moderate deviations

Rights: Copyright © 1982 Institute of Mathematical Statistics

Vol.10 • No. 2 • May, 1982
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