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.

We propose a novel Rayleigh quotient based sparse quadratic dimension reduction methodnamed QUADRO (Qua dratic D imension R eduction via Rayleigh O ptimization)for analyzing high-dimensional data. Unlike in the linear setting where Rayleigh quotient optimization coincides with classification, these two problems are very different under nonlinear settings. In this paper, we clarify this difference and show that Rayleigh quotient optimization may be of independent scientific interests. One major challenge of Rayleigh quotient optimization is that the variance of quadratic statistics involves all fourth cross-moments of predictors, which are infeasible to compute for high-dimensional applications and may accumulate too many stochastic errors. This issue is resolved by considering a family of elliptical models. Moreover, for heavy-tail distributions, robust estimates of mean vectors and covariance matrices are

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.

A semilinear in-slide model is introduced to remove the intensity effect in the scanning process. It is demonstrated that the intensity effect can be estimated accurately and removed effectively. This normalization step is vital for Affymetrix arrays to reveal relevant biological results when comparing gene expression in multiple arrays. The normalized expression ratios are analyzed further by a modified two-sample <i>t</i> test along with a sieved permutation scheme for computing <i>P</i> values. The improved specificity and sensitivity are demonstrated by using a study on the impact of macrophage migration inhibitory factor (MIF) reduction in neuroblastoma cells. With semilinear in-slide model analysis, expression of 166 genes was altered with a <i>P</i> value no greater than 0.001. Among those genes, 44 were altered &gt;2-fold. MIF-regulated genes associated with tumor development including IL-8 and <i>C</i>-<i>met</i>, which are overexpressed

Generalized linear models and quasi-likelihood method extend the ordinary regression models to accommodate more general conditional distributions of the response. Nonparametric methods need no explicit parametric specification and the resulting model is completely determined by the data themselves. However nonparametric estimation schemes generally have a slower convergence rate such as the local polynomial smoothing estimation of nonparametric generalized linear models studied in Fan, Heckman and Wand (1995). In this work, we propose two parametrically guided nonparametric estimation schemes by incorporating prior shape information on the link transformation of the response variables conditional mean in terms of the predictor variable. Asymptotic results and numerical simulations demonstrate the improvement of our new estimation schemes over the original nonparametric counterpart.