Statistica Sinica 13(2003), 275-282

STRONG GAUSSIAN APPROXIMATIONS IN THE

LEFT TRUNCATED AND RIGHT CENSORED MODEL

SzeMan Tse

National Donghua University

Abstract: In this paper, we consider the Tsai-Jewel-Wang estimator $F^0_n(x)$ of an unknown distribution function $F^0$ when the data are subject to random left truncation and right censorship. Strong Gaussian approximations of the product-limit process $\sqrt n [F_n^0(x)-F^0(x)]$ are constructed with rate $O({(\log n)^{3/2}/n^{1/8}})$. A functional law of the iterated logarithm for the maximal deviation of the estimator from the estimand is derived from the construction.

Key words and phrases: Cumulative hazard, left truncation, Gaussian approximations, right censorship, product-limit.