Consider a linear model Y= X + z, where X= X n, p and z~ N (0, I n). The vector is unknown and it is of interest to separate its nonzero coordinates from the zero ones (ie, variable selection). Motivated by examples in long-memory time series (Fan and Yao, 2003) and the change-point problem (Bhattacharya, 1994), we are primarily interested in the case where the Gram matrix G= X X is non-sparse but sparsifiable by a finite order linear filter. We focus on the regime where signals are both rare and weak so that successful variable selection is very challenging but is still possible.

Local quasi-likelihood estimation is useful for nonparametric modeling in a widely-used exponential family of distributions, called generalized linear models. Yet, the technique cannot be directly applied to situations where a response variable is missing at random. Three local quasi-likelihood estimation techniques are introduced: the local quasi-likelihood estimator using only complete-data; the locally weighted quasi-likelihood method; the local quasi-likelihood estimator with imputed values. These estimators share basically the same first order asymptotic biases and variances. Our simulation results show that substantial efficiency gains can be obtained by using the local quasi-likelihood estimator with imputed values. We develop the local quasi-likelihood imputation methods for estimating the mean functional of the response variable. It is shown that the proposed mean imputation estimators are asymptotically normal with asymptotic variance that can be easily estimated. Data from an ongoing environmental epidemiologic study is used to illustrate the proposed methods.

Discrepancy is a kind of important measure used in experimental design. Recently, a so-called discrete discrepancy has been applied to evaluate the uniformity of factorial designs. In this paper, we review some recent advances on application of the discrete discrepancy to several common experimental designs and summarize some important results.

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

In the analysis of cluster data the regression coefficients are frequently assumed to be the same across all clusters. This hampers the ability to study the varying impacts of factors on each cluster. In this paper, a semiparametric model is introduced to account for varying impacts of factors over clusters by using cluster-level covariates. It achieves the parsimony of parametrization and allows the explorations of nonlinear interactions. The random effect in the semiparametric model accounts also for within cluster correlation. Local linear based estimation procedure is proposed for estimating functional coefficients, residual variance, and within cluster correlation matrix. The asymptotic properties of the proposed estimators are established and the method for constructing simultaneous confidence bands are proposed and studied. In addition, relevant hypothesis testing problems are addressed. Simulation studies are