Nonparametric estimation and regression analysis with left-truncated and right-censored data.

*(English)*Zbl 0882.62037Summary: In many prospective and retrospective studies, survival data are subject to left truncation in addition to the usual right censoring. For left-truncated data without covariates, only the conditional distribution of the survival time \(Y\) given \(Y\geq \tau\) can be estimated nonparametrically, where \(\tau\) is the lower boundary of the support of the left-truncation variable \(T\). If the data are also right censored, then the conditional distribution can be consistently estimated only at points not larger than \(\tau^*\), where \(\tau^*\) is the upper boundary of the support of the right-censoring variable \(C\).

We first consider nonparametric estimation of trimmed functionals of the conditional distribution of \(Y\), with the trimming inside the observable range between \(\tau\) and \(\tau^*\). We then extend the approach to regression analysis and curve fitting in the presence of left truncation and right censoring on the response variable \(Y\). Asymptotic normality of M estimators of the regression parameters derived from this approach is established, and the result is used to construct confidence regions for the regression parameters. We also apply our methods of nonparametric estimation, correlation analysis, and curve fitting for left-truncated and right-censored data to analyze transfusion-induced AIDS data, and present a simulation study comparing our approach with another kind of M estimators for regression analysis in the presence of left truncation and right censoring.

We first consider nonparametric estimation of trimmed functionals of the conditional distribution of \(Y\), with the trimming inside the observable range between \(\tau\) and \(\tau^*\). We then extend the approach to regression analysis and curve fitting in the presence of left truncation and right censoring on the response variable \(Y\). Asymptotic normality of M estimators of the regression parameters derived from this approach is established, and the result is used to construct confidence regions for the regression parameters. We also apply our methods of nonparametric estimation, correlation analysis, and curve fitting for left-truncated and right-censored data to analyze transfusion-induced AIDS data, and present a simulation study comparing our approach with another kind of M estimators for regression analysis in the presence of left truncation and right censoring.

##### MSC:

62G07 | Density estimation |

62P10 | Applications of statistics to biology and medical sciences; meta analysis |