Semiparametric estimation of the hazard function in a model with covariate measurement error
Abstract
We consider a model where the failure hazard function, conditional on a covariate $Z$ is given by $R(t,\theta^0Z)=\eta\_{\gamma^0}(t)f\_{\beta^0}(Z)$, with $\theta^0=(\beta^0,\gamma^0)^\top\in \mathbb{R}^{m+p}$. The baseline hazard function $\eta\_{\gamma^0}$ and relative risk $f\_{\beta^0}$ belong both to parametric families. The covariate $Z$ is measured through the error model $U=Z+\epsilon$ where $\epsilon$ is independent from $Z$, with known density $f\_\epsilon$. We observe a $n$sample $(X\_i, D\_i, U\_i)$, $i=1,...,n$, where $X\_i$ is the minimum between the failure time and the censoring time, and $D\_i$ is the censoring indicator. We aim at estimating $\theta^0$ in presence of the unknown density $g$. Our estimation procedure based on least squares criterion provide two estimators. The first one minimizes an estimation of the least squares criterion where $g$ is estimated by density deconvolution. Its rate depends on the smoothnesses of $f\_\epsilon$ and $f\_\beta(z)$ as a function of $z$,. We derive sufficient conditions that ensure the $\sqrt{n}$consistency. The second estimator is constructed under conditions ensuring that the least squares criterion can be directly estimated with the parametric rate. These estimators, deeply studied through examples are in particular $\sqrt{n}$consistent and asymptotically Gaussian in the Cox model and in the excess risk model, whatever is $f\_\epsilon$.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 2006
 arXiv:
 arXiv:math/0606192
 Bibcode:
 2006math......6192M
 Keywords:

 Mathematics  Statistics;
 62G05;
 62F12;
 62N01;
 62N02;
 62J02