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

Abstract

We consider a failure hazard function,conditional on a time-independent covariate Z,given by $\eta_{\gamma^0}(t)f_{\beta^0}(Z)$ . The baseline hazardfunction $\eta_{\gamma^0}$ and the relative risk $f_{\beta^0}$ both belong to parametricfamilies 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 anadditive error model U = Z + ε where ε is a random variable, independent from Z, withknown density $f_\varepsilon$ .We observe a n-sample (X_i, D_i, U_i), i = 1, ..., n, where X _i isthe 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 θ ⁰ using the observationsn-sample (X_i, D_i, U_i), i = 1, ..., n.
We give an upper bound for its riskwhich depends on the smoothness properties of $f_\varepsilon$ and $f_\beta(z)$ as afunction of z, and we derive sufficient conditionsfor the $\sqrt{n}$ -consistency.We give detailed examples consideringvarious type of relative risks $f_{\beta}$ and various types of errordensity $f_\varepsilon$ . In particular, in the Cox model and inthe excess risk model, the estimator of θ ⁰ is $\sqrt{n}$ -consistent and asymptotically Gaussianregardless of the form of $f_\varepsilon$ .