Parametric conditional variance estimation in location-scale models with censored data

Abstract

Suppose the random vector $(X,Y)$ satisfies the regression model $Y=m(X)+\sigma (X)\varepsilon$, where $m(\cdot)=E(Y|\cdot),$ $\sigma^{2}(\cdot)=\mbox{Var}(Y|\cdot)$ belongs to some parametric class $\{\sigma _{\theta}(\cdot):\theta \in \Theta\}$ and $\varepsilon$ is independent of $X$. The response $Y$ is subject to random right censoring and the covariate $X$ is completely observed. A new estimation procedure is proposed for $\sigma _{\theta}(\cdot)$ when $m(\cdot)$ is unknown. It is based on nonlinear least squares estimation extended to conditional variance in the censored case. The consistency and asymptotic normality of the proposed estimator are established. The estimator is studied via simulations and an important application is devoted to fatigue life data analysis.