The advance of technology facilitates the collection of statistical data. Flexible and refined statistical models are widely sought in a large array of statistical problems. The question arises frequently whether or not a family of parametric or nonparametric models fit adequately the given data. In this paper we give a selective overview on nonparametric inferences using generalized likelihood ratio (<i>GLR</i>) statistics. We introduce generalized likelihood ratio statistics to test various null hypotheses against nonparametric alternatives. The trade-off between the flexibility of alternative models and the power of the statistical tests is emphasized. Well-established Wilks phenomena are discussed for a variety of semi- and non-parametric models, which sheds light on other research using <i>GLR</i> tests. Anumber of open topics worthy of further study are given in a discussion section.

The quantitative comparison of two or more microarrays can reveal, for example, the distinct patterns of gene expression that define different cellular phenotypes or the genes that are induced in the cellular response to certain stimulations. Normalization of the measured intensities is a prerequisite of such comparisons. However, a fundamental problem in cDNA microarray analysis is the lack of a common standard to compare the expression levels of different samples. Several normalization protocols have been proposed to overcome the variabilities inherent in this technology. We have developed a normalization procedure based on within-array replications via a semilinear in-slide model, which adjusts objectively experimental variations without making critical biological assumptions. The significant analysis of gene expressions is based on a weighted <i>t</i> statistic, which accounts for the heteroscedasticity of the

High throughput genetic sequencing arrays with thousands of measurements per sample and a great amount of related censored clinical data have increased demanding need for better measurement specific model selection. In this paper we establish strong oracle properties of non-concave penalized methods for non-polynomial (NP) dimensional data with censoring in the framework of Coxs proportional hazards model. A class of folded-concave penalties are employed and both LASSO and SCAD are discussed specifically. We unveil the question under which dimensionality and correlation restrictions can an oracle estimator be constructed and grasped. It is demonstrated that non-concave penalties lead to significant reduction of the irrepresentable condition needed for LASSO model selection consistency. The large deviation result for martingales, bearing interests of its own, is developed for characterizing

We discuss the difficulties of estimating quadratic functionals based on observations Y (t) from the white noise model Y (t) where Y (t) is a standard Wiener process on Y (t) . The optimal rates of convergence (as Y (t) ) for estimating quadratic functionals under certain geometric constraints are found. Specifically, the optimal rates of estimating Y (t) under hyperrectangular constraints Y (t) and weighted Y (t) -body constraints Y (t) are computed explicitly, where Y (t) is the Y (t) th Fourier-Bessel coefficient of the unknown function Y (t) . We develop lower bounds based on testing two highly composite hypercubes and address their advantages. The attainable lower bounds are found by applying the hardest one-dimensional approach as well as the hypercube method. We demonstrate that for estimating regular quadratic functionals [ie, the functionals which can be estimated

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.