Statistica Sinica 10(2000), 281-296

STRONG GAUSSIAN APPROXIMATIONS

IN THE RANDOM TRUNCATION MODE

SzeMan Tse

National Donghua University

Abstract: In the random left-truncation model, one observes $(X_i,Y_i)$ only if $X_i \ge Y_i$, $i=1$, $\ldots \,$, $N$. The nonparametric maximum likelihood estimator aims at reconstructing the distribution function of $X$ from the observed empirical data. In this paper, strong approximations of the cumulative hazard process and product-limit process on increasing sets by sequences of copies of corresponding Gaussian limiting processes are constructed. The convergence rates are $N^{-1/6}\log N$ on fixed sets. Futhermore, strong approximations with two-parameter Gaussian processes are obtained with convergence rates $N^{-1/8}(\log N)^{3/2}$ on fixed sets.

Key words and phrases: Cumulative hazard, Gaussian approximations, product-limit, random truncation.