Bahadur-Kiefer theorems for the product-limit process
Suppose that \((X_ i)\) is a sequence of i.i.d. random variables that is censored from the right by an i.i.d. process \((Y_ i)\) so that only \(Z_ i=\min (X_ i,Y_ i)\) and \(\delta_ i=1_{\{X_ i\leq Y_ i\}}\) are observable. The product-limit (PL) estimator (Kaplan-Meier estimator) \(F_ n(x)\) of \(F(x)=P(X_ 1\leq x)\) is given by \[ 1-F_ n(x)=\prod_{Z_{(i)}\leq x}(1-\delta_{(i)}/(n-i+1)). \] A natural estimator of the quantile function \(Q(t)=F^{-1}(t)\) is the PL quantile estimator \(Q_ n(t)=F_ n^{-1}(t).\) The authors study the PL process \(\alpha_ n(x)=n^{1/2}(F_ n(x)- F(x))\) and the PL quantile process \[ \beta_ n(t)=n^{1/2} f(Q(t))(Q_ n(t)-Q(t)), \] where \(f=F'\). For the remainder term \(R_ n(t)=\alpha_ n(Q(t))+\beta_ n(t)\) in the Bahadur-Kiefer representation they first prove finite-dimensional convergence. Then they establish an almost sure upper bound at a fixed point. The main result is a precise limit theorem in the supremum norm.
scientific article; zbMATH DE number 4172167