I.i.d. representations for the bivariate product limit estimators and the bootstrap versions. (English) Zbl 0678.62043
Summary: Let \(F(s,t)=P(X>s\), \(Y>t)\) be the bivariate survival function which is subject to random censoring. Let \(\hat F_ n(s,t)\) be the bivariate product limit estimator (PL-estimator) by G. Campbell and A. Földes [Nonparametric statistical inference, Budapest 1980, Vol. I, Colloq. Math. Soc. Janos Bolyai 32, 103-121 (1982; Zbl 0516.62041)]. In this paper, it was shown that
\[
\hat F_ n(s,t)-F(s,t)=n^{- 1}\sum^{n}_{i=1}\zeta_ i(s,t)+R_ n(s,t),
\]
where \(\{\zeta_ i(s,t)\}\) 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 \(\{n^{-1}\sum^{n}_{i=1}\zeta_ i(s,t)\}\) 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 results can be extended to the multivariate cases and are extensions of the univariate case of S.-H. Lo and K. Singh [Probab. Theory Relat. Fields 71, 455-465 (1986; Zbl 0561.62032)]. The estimator \(\hat F_ n(s,t)\) is also modified so that the modified estimator is closer to the true survival function than \(\hat F_ n(s,t)\) in sup norm.
The results can be extended to the multivariate cases and are extensions of the univariate case of S.-H. Lo and K. Singh [Probab. Theory Relat. Fields 71, 455-465 (1986; Zbl 0561.62032)]. The estimator \(\hat F_ n(s,t)\) is also modified so that the modified estimator is closer to the true survival function than \(\hat F_ n(s,t)\) in sup norm.
MSC:
| 62G05 | Nonparametric estimation |
| 62E20 | Asymptotic distribution theory in statistics |
| 60F05 | Central limit and other weak theorems |
| 60F17 | Functional limit theorems; invariance principles |
Keywords:
functional weak convergence; function LIL; bivariate survival function; random censoring; bivariate product limit estimator; two-dimensional-time Gaussian process; covariance structure; limiting Gaussian process; bootstrap estimatorsReferences:
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