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July, 1992 Functional Laws of the Iterated Logarithm for the Increments of Empirical and Quantile Processes
Paul Deheuvels, David M. Mason
Ann. Probab. 20(3): 1248-1287 (July, 1992). DOI: 10.1214/aop/1176989691


Let $\{\alpha_n(t), 0 \leq t \leq 1\}$ and $\{\beta_n(t), 0 \leq t \leq 1\}$ be the empirical and quantile processes generated by the first $n$ observations from an i.i.d. sequence of uniformly distributed random variables on (0,1). Let $0 < a_n < 1$ be a sequence of constants such that $a_n \rightarrow 0$ as $n \rightarrow \infty$. We investigate the strong limiting behavior as $n \rightarrow \infty$ of the increment functions $\{\alpha_n(t + a_ns) - \alpha_n(t), 0 \leq s \leq 1\}$ and $\{\beta_n(t + a_ns) - \beta_n(t), 0 \leq s \leq 1\},$ where $0 \leq t \leq 1 - a_n$. Under suitable regularity assumptions imposed upon $a_n$, we prove functional laws of the iterated logarithm for these increment functions and discuss statistical applications in the field of nonparametric estimation.


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Paul Deheuvels. David M. Mason. "Functional Laws of the Iterated Logarithm for the Increments of Empirical and Quantile Processes." Ann. Probab. 20 (3) 1248 - 1287, July, 1992.


Published: July, 1992
First available in Project Euclid: 19 April 2007

zbMATH: 0767.60028
MathSciNet: MR1175262
Digital Object Identifier: 10.1214/aop/1176989691

Primary: 60F15
Secondary: 60F17 , 62G05

Keywords: Density estimation , Empirical processes , Functional limit laws , Laws of the iterated logarithm , nearest neighbor estimates , nonparametric estimation , order statistics , Quantile processes

Rights: Copyright © 1992 Institute of Mathematical Statistics


Vol.20 • No. 3 • July, 1992
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