The non-Gaussian maximum likelihood estimator is frequently used in GARCH models with the intention of capturing heavy-tailed returns. However, unless the parametric likelihood family contains the true likelihood, the estimator is inconsistent due to density misspecification. To correct this bias, we identify an unknown scale parameter _<i>f</i> that is critical to the identification for consistency and propose a three-step quasi-maximum likelihood procedure with non-Gaussian likelihood functions. This novel approach is consistent and asymptotically normal under weak moment conditions. Moreover, it achieves better efficiency than the Gaussian alternative, particularly when the innovation error has heavy tails. We also summarize and compare the values of the scale parameter and the asymptotic efficiency for estimators based on different choices of likelihood functions with an increasing level of heaviness in the

In this paper, we propose a penalised pseudo-partial likelihood method for variable selection with multivariate failure time data with a growing number of regression coefficients. Under certain regularity conditions, we show the consistency and asymptotic normality of the penalised likelihood estimators. We further demonstrate that, for certain penalty functions with proper choices of regularisation parameters, the resulting estimator can correctly identify the true model, as if it were known in advance. Based on a simple approximation of the penalty function, the proposed method can be easily carried out with the NewtonRaphson algorithm. We conduct extensive Monte Carlo simulation studies to assess the finite sample performance of the proposed procedures. We illustrate the proposed method by analysing a dataset from the Framingham Heart Study.

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