Semi-parametric 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 $R(t,\theta^0|Z)=\eta_{\gamma^0}(t)f_{\beta^0}(Z)$, 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 $U=Z+\varepsilon$ where $\varepsilon$ is independent from $Z$, with known density $f_\varepsilon$. We observe a $n$-sample $(X_i, D_i, U_i)$, $i=1,\ldots,n$, 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 $f_\varepsilon$ 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 $f_\varepsilon$.
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Pré-publication, Document de travail
2006
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https://hal.archives-ouvertes.fr/hal-00079007
Contributeur : Marie-Luce Taupin <>
Soumis le : jeudi 8 juin 2006 - 16:07:34
Dernière modification le : jeudi 9 février 2017 - 15:55:02
Document(s) archivé(s) le : lundi 5 avril 2010 - 22:31:22

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Marie-Laure Martin-Magniette, Marie-Luce Taupin. Semi-parametric estimation of the hazard function in a model with covariate measurement error. 2006. <hal-00079007>

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