The Annals of Probability

Functional Laws of the Iterated Logarithm for the Product-Limit Estimator of a Distribution Function Under Random Censorship or Truncation

Ming Gao Gu and Tze Leung Lai

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Abstract

Functional laws of the iterated logarithm are established for a modified version of the classical product-limit estimator of a distribution function when the data are subject to random censorship or truncation. These functional laws are shown to hold for the entire interval $I$ over which the distribution function can be consistently estimated, under basically the same assumptions that have been used in the literature to establish the weak convergence of the normalized estimator in $D(I)$. Making use of stochastic integral representations and empirical process theory, strong approximations involving i.i.d. continuous-parameter martingales are developed for the product-limit estimator, and these strong approximations are then applied to derive the functional laws of the iterated logarithm.

Article information

Source
Ann. Probab. Volume 18, Number 1 (1990), 160-189.

Dates
First available: 19 April 2007

Permanent link to this document
http://projecteuclid.org/euclid.aop/1176990943

JSTOR
links.jstor.org

Digital Object Identifier
doi:10.1214/aop/1176990943

Mathematical Reviews number (MathSciNet)
MR1043942

Zentralblatt MATH identifier
0705.62040

Subjects
Primary: 60F15: Strong theorems
Secondary: 60F17: Functional limit theorems; invariance principles 60H05: Stochastic integrals 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case) 62G05: Estimation

Keywords
Functional LIL strong approximation martingales stochastic integrals empirical processes product-limit estimator reproducing kernel Hilbert space

Citation

Gu, Ming Gao; Lai, Tze Leung. Functional Laws of the Iterated Logarithm for the Product-Limit Estimator of a Distribution Function Under Random Censorship or Truncation. The Annals of Probability 18 (1990), no. 1, 160--189. doi:10.1214/aop/1176990943. http://projecteuclid.org/euclid.aop/1176990943.


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