Hazard estimation with censoring and measurement error: application to length of pregnancy

Estimation of the physiological length of pregnancy is a challenging problem since both the time origin of the pregnancy and the time of onset of labor are partly observed. The time to spontaneous labor is indeed right-censored, and the time of fertilization is only known up to an error. Therefore, data are subject to both censoring and measurement errors. We focus on the case where the measurement errors affect both the variable of interest and the censoring variable, which is the case of the timing of spontaneous delivery among pregnant women. We propose an estimation strategy to estimate the hazard rate of the underlying variable of interest. We explain the model and this strategy and provide L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-risk bound for the data driven resulting estimator. We also derive estimators of the survival function and the density. Simulations illustrate the performances of the estimator. Lastly, the method is applied to an original real data set of length of pregnancy to estimate rates of previable births, severe preterm births and prolonged pregnancy and the influence of the cervical length of the first semester.