In survival analysis, the relationship between a survival time and a covariate is conveniently modeled with the proportional hazards regression model. This model usually assumes that the covariate has a log-linear effect on the hazard function. In this paper we consider the proportional hazards regression model with a nonparametric risk effect. We discuss estimation of the risk function and its derivatives in two cases: when the baseline hazard function is parametrized and when it is not parametrized. In the case of a parametric baseline hazard function, inference is based on a local version of the likelihood function, while in the case of a nonparametric baseline hazard, we use a local version of the partial likelihood. This results in maximum local likelihood estimators and maximum local partial likelihood estimators, respectively. We establish the asymptotic normality of the estimators. It turns out that both methods have

This paper considers a proportional hazards model, which allows one to examine the extent to which covariates interact nonlinearly with an exposure variable, for analysis of lifetime data. A local partial-likelihood technique is proposed to estimate nonlinear interactions. Asymptotic normality of the proposed estimator is established. The baseline hazard function, the bias and the variance of the local likelihood estimator are consistently estimated. In addition, a one-step local partial-likelihood estimator is presented to facilitate the computation of the proposed procedure and is demonstrated to be as efficient as the fully iterated local partial-likelihood estimator. Furthermore, a penalized local likelihood estimator is proposed to select important risk variables in the model. Numerical examples are used to illustrate the effectiveness of the proposed procedures.

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Quadratic regression functionals are important for bandwidth selection of nonparametric regression techniques and for nonparametric goodness-of-fit test. Based on local polynomial regression, we propose estimators for weighted integrals of squared derivatives of regression functions. The rates of convergence in mean square error are calculated under various degrees of smoothness and appropriate values of the smoothing parameter. Asymptotic distributions of the proposed quadratic estimators are considered with the Gaussian noise assumption. It is shown that when the estimators are pseudo-quadratic (linear components dominate quadratic components), asymptotic normality with rate n-1/2 can be achieved.

We study the estimation of the additive components in additive regression models, based on the weighted sample average of regression surface, for stationary <i></i>-mixing processes. Explicit expression of this method makes possible a fast computation and allows an asymptotic analysis. The estimation procedure is especially useful for additive modeling. In this paper, it is shown that the average surface estimator shares the same optimality as the ideal estimator and has the same ability to estimate the additive component as the ideal case where other components are known. Formulas for the asymptotic bias and normality of the estimator are established. A small simulation study is carried out to illustrate the performance of the estimation and a real example is also used to demonstrate our methodology.