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

Marie-Laure Martin-Magniette; Marie-Luce Taupin

ESAIM: Probability and Statistics (2009)

  • Volume: 13, page 87-114
  • ISSN: 1292-8100

Abstract

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We consider a failure hazard function, conditional on a time-independent covariate Z, given by η γ 0 ( t ) f β 0 ( Z ) . The baseline hazard function η γ 0 and the relative risk f β 0 both belong to parametric families with θ 0 = ( β 0 , γ 0 ) ⊤ ∈ ℝ m + p . The covariate Z has an unknown density and is measured with an error through an additive error model U = Z + ε where ε is a random variable, independent from Z, with known density f ε . We observe a n-sample (Xi, Di, Ui), i = 1, ..., n, where Xi is the minimum between the failure time and the censoring time, and Di is the censoring indicator. Using least square criterion and deconvolution methods, we propose a consistent estimator of θ0 using the observations n-sample (Xi, Di, Ui), i = 1, ..., n.
We give an upper bound for its risk which depends on the smoothness properties of f ε and f β ( z ) as a function of z, and we derive sufficient conditions for the n -consistency. We give detailed examples considering various type of relative risks f β and various types of error density f ε . In particular, in the Cox model and in the excess risk model, the estimator of θ0 is n -consistent and asymptotically Gaussian regardless of the form of f ε .

How to cite

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Martin-Magniette, Marie-Laure, and Taupin, Marie-Luce. "Estimation of the hazard function in a semiparametric model with covariate measurement error." ESAIM: Probability and Statistics 13 (2009): 87-114. <http://eudml.org/doc/250633>.

@article{Martin2009,
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 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 + ε where ε is a random variable, independent from Z, with known density $f_\varepsilon$. We observe a n-sample (Xi, Di, Ui), i = 1, ..., n, where Xi is the minimum between the failure time and the censoring time, and Di is the censoring indicator. Using least square criterion and deconvolution methods, we propose a consistent estimator of θ0 using the observations n-sample (Xi, Di, Ui), i = 1, ..., n.
We give an upper bound for its risk which depends on the smoothness properties of $f_\varepsilon$ and $f_\beta(z)$ 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 $f_\varepsilon$. In particular, in the Cox model and in the excess risk model, the estimator of θ0 is $\sqrt\{n\}$-consistent and asymptotically Gaussian regardless of the form of $f_\varepsilon$. },
author = {Martin-Magniette, Marie-Laure, Taupin, Marie-Luce},
journal = {ESAIM: Probability and Statistics},
keywords = {Semiparametric estimation; errors-in-variables model; measurement error; nonparametric estimation; excess risk model; Cox model; censoring; survival analysis; density deconvolution; least square criterion; semiparametric estimation; least squares criterion},
language = {eng},
month = {3},
pages = {87-114},
publisher = {EDP Sciences},
title = {Estimation of the hazard function in a semiparametric model with covariate measurement error},
url = {http://eudml.org/doc/250633},
volume = {13},
year = {2009},
}

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
DA - 2009/3//
PB - EDP Sciences
VL - 13
SP - 87
EP - 114
AB - 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 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 + ε where ε is a random variable, independent from Z, with known density $f_\varepsilon$. We observe a n-sample (Xi, Di, Ui), i = 1, ..., n, where Xi is the minimum between the failure time and the censoring time, and Di is the censoring indicator. Using least square criterion and deconvolution methods, we propose a consistent estimator of θ0 using the observations n-sample (Xi, Di, Ui), i = 1, ..., n.
We give an upper bound for its risk which depends on the smoothness properties of $f_\varepsilon$ and $f_\beta(z)$ 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 $f_\varepsilon$. In particular, in the Cox model and in the excess risk model, the estimator of θ0 is $\sqrt{n}$-consistent and asymptotically Gaussian regardless of the form of $f_\varepsilon$.
LA - eng
KW - Semiparametric estimation; errors-in-variables model; measurement error; nonparametric estimation; excess risk model; Cox model; censoring; survival analysis; density deconvolution; least square criterion; semiparametric estimation; least squares criterion
UR - http://eudml.org/doc/250633
ER -

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