Estimation of the hazard function in a semiparametric model with covariate measurement error

Martin-Magniette, Marie-Laure; Taupin, Marie-Luce

ESAIM: Probability and Statistics, Tome 13 (2009), pp. 87-114.

We consider a failure hazard function, conditional on a time-independent covariate $Z$, given by ${\eta }_{{\gamma }^0}\left(t\right)f_{{\beta }^0}\left(Z\right)$. The baseline hazard function ${\eta }_{{\gamma }^0}$ and the relative risk $f_{{\beta }^0}$ both belong to parametric families with ${\theta }^0={({\beta }^0,{\gamma }^0)}^{\top }\in {ℝ}^{m+p}$. The covariate $Z$ has an unknown density and is measured with an error through an additive error model $U=Z+\epsilon$ where $\epsilon$ is a random variable, 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. Using least square criterion and deconvolution methods, we propose a consistent estimator of ${\theta }^0$ using the observations $(X_i,D_i,U_i)$, $i$ = 1, ..., $n$. We give an upper bound for its risk which depends on the smoothness properties of $f_{\epsilon }$ and $f_{\beta }\left(z\right)$ as a function of $z$, and we derive sufficient conditions for the $\sqrt{n}$-consistency. We give detailed examples considering various type of relative risks $f_{\beta }$ and various types of error density $f_{\epsilon }$. In particular, in the Cox model and in the excess risk model, the estimator of ${\theta }^0$ is $\sqrt{n}$-consistent and asymptotically gaussian regardless of the form of $f_{\epsilon }$.

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