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

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.