I.i.d. representations for the bivariate product limit estimators and the bootstrap versions

Abstract

Let F( s, t) = P( X > s, Y > t) be the bivariate survival function which is subject to random censoring. Let F ̂ n(s, t) be the bivariate product limit estimator (PL-estimator) by Campbell and Földes (1982, Proceedings International Colloquium on Non-parametric Statistical Inference, Budapest 1980, North-Holland, Amsterdam). In this paper, it was shown that F ̂ n(s, t) − F(s, t) = n −1 Σ i = 1 n ζ i(s, t) + R n(s, t) , where { ζ i ( s, t)} is i.i.d. mean zero process and R n ( s, t) is of the order O((n −1 log n) 3 4 ) a.s. uniformly on compact sets. Weak convergence of the process { n −1 Σ i = 1 n ζ i ( s, t)} to a two-dimensional-time Gaussian process is shown. The covariance structure of the limiting Gaussian process is also given. Corresponding results are also derived for the bootstrap estimators. The result can be extended to the multivariate cases and are extensions of the univariate case of Lo and Singh (1986, Probab. Theory Relat. Fields, 71, 455–465). The estimator F ̂ n(s, t) is also modified so that the modified estimator is closer to the true survival function than F ̂ n(s, t) in supnorm.