Kaplan-Meier estimator under association
This work studies the Kaplan-Meier estimator and the estimation of the hazard function in a model with censored failure times. The true survival times \(T_1,\dots T_n\), with common marginal \(F\), are not assumed mutually independent. They satisfy two different notions of weak dependence: a) They are positively associated, i.e. \[ \text{cov}[G(X_i;1\leq i\leq n),H(X_j;1\leq j\leq n)]\geq 0\;\text{for} E[G^2(X_j;1 \leq j \leq n)]<\infty \] \[ \text{and} E[H^2(X_j; 1 \leq j\leq n)]<\infty,\;G,H : R^n\rightarrow R \] and are coordinatewise nondecreasing; or b) negatively associated, i.e., \[ \text{cov}[G(X_i; i\in A),H (X_j; j\in A^c)]\leq 0. \] \(A\) is a proper subset of \(\{1\dots, n\}\), \(G : R^{\# A}\rightarrow R\) and \(H : R^{\# A^c}\rightarrow R\) are functions with the same properties as above. The censored r.v.s \(\{Y_i\}_{i=1}^{n}\) are assumed i.i.d. and independent of the \(T_i\)'s. Defining \[ N_n(t)=\sum_{i=1}^{n}I(T_i\wedge Y_i)\leq t, I(T_i\wedge Y_i)=1)\;\text{and} Y_n(t)=\sum_{i=1}^{n}I(T_i\wedge Y_i)\geq t). \] The Kaplan-Meier estimator of \(F\) is always defined as \(1-\tilde F_n(t)=\prod_{s\leq t}(1-\frac{dN_n(s)}{Y_n(s)})\) where \(dN_n(s)=N_n(s)-N_n(s-)\). The hazard function \(\Delta(t)\) is given by \(\Delta(t)=\int_0^t\frac{dF(s)}{1-F(s-)}\) and its empirical estimator is \(\hat \Delta_n(t)=\int_0^t\frac{dN_n(s)}{Y_n(s)}\). Under some conditions the authors obtain several results that are generalizations of the i.i.d. case, namely: 1) \(sup_{0\leq t\leq \tau}|\hat \Delta_n(t)-\Delta(t)|\rightarrow 0 \) a.s. under the two types of dependence, 2) \(sup_{0\leq t\leq \tau_H}|\tilde F_n(t)-F(t)|\rightarrow 0 \) a.s. and \(sup_{0\leq t\leq Z_{n:n}}|\tilde F_n(t)-F(t)|\rightarrow 0 \) a.s. where \(Z_{n:n}=max_{i\leq n}(T_i\wedge Y_i)\), rates in these two convergences are also obtained. Finally in Theorem 1.5 they prove a weak convergence result, towards a Gaussian process, for the empirical process \(\tilde Z_n(t)=\sqrt n \frac{\tilde F_n(t)-F(t)}{1-F(t)}\). The work ends with a construction of a variance estimator for the variance of the Gaussian process limit.
scientific article; zbMATH DE number 1241228