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 ${\theta}^{0}={({\beta}^{0},{\gamma}^{0})}^{\top}\in {\mathbb{R}}^{m+p}$. 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 ${f}_{\epsilon}$.

Keywords: semiparametric estimation, errors-in-variables model, measurement error, nonparametric estimation, excess risk model, Cox model, censoring, survival analysis, density deconvolution, least square criterion

@article{PS_2009__13__87_0, author = {Martin-Magniette, Marie-Laure and Taupin, Marie-Luce}, title = {Estimation of the hazard function in a semiparametric model with covariate measurement error}, journal = {ESAIM: Probability and Statistics}, pages = {87--114}, publisher = {EDP-Sciences}, volume = {13}, year = {2009}, doi = {10.1051/ps:2008004}, mrnumber = {2502025}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/ps:2008004/} }

TY - JOUR AU - Martin-Magniette, Marie-Laure AU - Taupin, Marie-Luce TI - Estimation of the hazard function in a semiparametric model with covariate measurement error JO - ESAIM: Probability and Statistics PY - 2009 SP - 87 EP - 114 VL - 13 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ps:2008004/ DO - 10.1051/ps:2008004 LA - en ID - PS_2009__13__87_0 ER -

%0 Journal Article %A Martin-Magniette, Marie-Laure %A Taupin, Marie-Luce %T Estimation of the hazard function in a semiparametric model with covariate measurement error %J ESAIM: Probability and Statistics %D 2009 %P 87-114 %V 13 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ps:2008004/ %R 10.1051/ps:2008004 %G en %F PS_2009__13__87_0

Martin-Magniette, Marie-Laure; Taupin, Marie-Luce. Estimation of the hazard function in a semiparametric model with covariate measurement error. ESAIM: Probability and Statistics, Volume 13 (2009), pp. 87-114. doi : 10.1051/ps:2008004. http://archive.numdam.org/articles/10.1051/ps:2008004/

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