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Density and hazard estimation in censored regression models. (English) Zbl 1007.62029
Summary: Let $$(X,Y)$$ be a random vector, where $$Y$$ denotes the variable of interest, possibly subject to random right censoring, and $$X$$ is a covariate. Consider a heteroscedastic model $$Y=m(X)+ \sigma(X)+ \sigma(X) \varepsilon$$, where the error term $$\varepsilon$$ is independent of $$X$$ and $$m(X)$$ and $$\sigma(X)$$ are smooth but unknown functions. Under this model, we construct a nonparametric estimator for the density and hazard function of $$Y$$ given $$X$$, which has a faster rate of convergence than the completely nonparametric estimator that is constructed without making any model assumption. Moreover, the proposed estimator for the density and hazard function performs better than the classical nonparametric estimator, especially in the right tail of the distribution.
We prove the weak convergence of both the density and the hazard function estimator. The results are obtained by constructing asymptotic representations for the two estimators and by making use of work by I. Van Keilegom and M. G. Akritas [Ann. Stat. 27, No. 5, 1745-1784 (1999; Zbl 0957.62034)] in which an estimator of the conditional distribution of $$Y$$ given $$X$$ is studied under the same model assumption.

##### MSC:
 62G07 Density estimation 62G08 Nonparametric regression and quantile regression 62G20 Asymptotic properties of nonparametric inference