Improving efficiency for regression coefficients and predicting trajectories of individuals are two important aspects in the analysis of longitudinal data. Both involve estimation of the covariance function. Yet challenges arise in estimating the covariance function of longitudinal data collected at irregular time points. A class of semiparametric models for the covariance function by that imposes a parametric correlation structure while allowing a nonparametric variance function is proposed. A kernel estimator for estimating the nonparametric variance function is developed. Two methods for estimating parameters in the correlation structurea quasi-likelihood approach and a minimum generalized variance methodare proposed. A semiparametric varying coefficient partially linear model for longitudinal data is introduced, and an estimation procedure for model coefficients using a profile weighted least squares approach

Variable selection is vital to statistical data analyses. Many of procedures in use are ad hoc stepwise selection procedures, which are computationally expensiveandignore stochastic errors in the variable selection process of previous steps. An automatic and simultaneous variable selection procedure can be obtained by using a penalized likelihood method. In traditional linear models, the best subset selection and stepwise deletion methods coincide with a penalized leastsquares method when design matrices are orthonormal. In this paper, we propose a few new approaches to selecting variables for linear models, robust regression models and generalized linear models based on a penalized likelihood approach. A family of thresholding functions are proposed. The LASSO proposed by Tibshirani (1996) is a member of the penalized leastsquares with the # #-penalty. A smoothly clipped absolute deviation (SCAD) penalty function is introduced to ameliorate the properties of # #-penalty. A unied algorithm is introduced, which is backed up by statistical theory. The new approaches are compared with the ordinary leastsquares methods, the garrote method by Breiman (1995) and the LASSO method by Tibshirani (1996). Our simulation results show that the newly proposed methods compare favorably with other approaches as an automatic variable selection technique. Because of simultaneous selection of variables and estimation of parameters, we are able to give a simple estimated standard error formula, which is tested to be accurate enough for practical applications. Two real data examples illustrate the versatility and eectiveness of the

I would like to congratulate Professor Antoniadis for successfully outlining the current state-of-art of wavelet applications in statistics. Since wavelet techniques were introduced to statistics in the early 90s, the applications of wavelet techniques have mushroomed. There is a vast forest of wavelet theory and techniques in statistics and one can find himself easily lost in the jungle. The review by Antoniadis, ranging from linear wavelets to nonlinear wavelets, addressing both theoretical issues and practical relevance, gives in-depth coverage of the wavelet applications in statistics and guides one entering easily into the realm of wavelets.

Motivated by recent work on studying massive imaging data in various neuroimaging studies, we propose a novel spatially varying coefficient model (SVCM) to capture the varying association between imaging measures in a three-dimensional volume (or two-dimensional surface) with a set of covariates. Two stylized features of neuorimaging data are the presence of multiple piecewise smooth regions with unknown edges and jumps and substantial spatial correlations. To specifically account for these two features, SVCM includes a measurement model with multiple varying coefficient functions, a jumping surface model for each varying coefficient function, and a functional principal component model. We develop a three-stage estimation procedure to simultaneously estimate the varying coefficient functions and the spatial correlations. The estimation procedure includes a fast multiscale adaptive estimation and

In this paper, we study, in some new ways, the estimation of unimodal densities. Several methods for estimating unimodal densities are proposed: plug-in MLE, pregrouping techniques, linear spline MLE. Based on the maximum likelihood method, an automatic procedure for estimating a unimodal density as well as its mode is proposed. We also give asymptotic theory for the proposed estimators. An important consequence of this study is that having to estimate the location of a mode does not affect the limiting behavior of the proposed unimodal density estimate. Simulation studies illustrate the proposed methods.