Uniform convergence of nonparametric regressions in competing risk models with right censoring

Abstract

a b s t r a c tWe consider, in the presence of covariates, non-independent competing risks that are subject to right censoring. We define a nonparametric estimator of the incident regression function through the generalized product-limit estimator of the conditional censorship distribution function. Under suitable conditions, we establish the almost sure uniform convergence of those estimators with an appropriate rate.

Nonparametric estimation of cumulative incidence functions for competing risks data with missing cause of failure

“…The convergence rate is almost identical to the rate of Bordes and Gneyou (2011) who study the uniform convergence rate just for right-censored competing risks data. In their result, the variance term is of the same order whereas their bias term goes faster to zero as it is of order h 2d .…”

“…The proposed nonparametric estimator is complementary to i) the developed (semi)-parametric procedures with right-censored data and continuous explanatory covariates (e.g., Andersen et al, 1993;Jeong and Fine, 2007;Scheike et al, 2008) and ii) the suggested parametric methods for right-censored data where the cause of failure is sometimes missing (Lu and Liang, 2008). Finally, we compare our results on uniform convergence rates with the results of Bordes and Gneyou (2011) who discuss the uniform convergence rate for the nonparametric estimator with right-censored competing risks data.…”

Quantifying and estimating the predictive accuracy for censored time‐to‐event data with competing risks

“…The asymptotic bias of two conditional survival probabilities as a function of bandwidth are in the same direction. 28 Therefore, the bias of their ratio can be canceled out to some extent, particularly when Ti and τ are close.…”

“…The numerator of W 1 i can be expressed as the cause‐specific survival probability between and τ : , and the denumerator is the overall survival probability beyond . The asymptotic bias of 2 conditional survival probabilities as a function of bandwidth are in the same direction . Therefore, the bias of their ratio can be canceled out to some extent, particularly when and τ are close.…”

A strong linear representation for the maximum conditional hazard rate estimator in survival analysis

“…and (a n ) n∈N are two non increasing sequences of positive real numbers such that (H1) (i) h n → 0, convergence of the estimators H n (t|z) and H 1n (t|z) to H(t|z) and H 1 (t|z) respectively as in [2] while assumptions (F2), (F3), (K2) and (H2) ensure the strong uniform convergence of λ n (t|z) and θ n (z) to λ(t|z) and θ (z) respectively.…”