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

Two measures of sensitivity to initial conditions in nonlinear time series are proposed. The notions give some insight into the relationship between the Fisher information in statistical estimation and initial-value sensitivity in dynamical systems. By using the locally polynomial regression, we develop nonparametric estimates for a conditional density function, its square root and its partial derivatives. The proposed procedures are innovative and of interests in their own right. They are also used to estimate the sensitive measures. The asymptotic normality of the proposed estimators have been proved. We also propose a simple and intuitively appealing method for choosing the bandwidths. Two simulated examples are used as illustrations.

We examine the effect of errors in covariates in nonparametric function estimation. These functions include densities, regressions and conditional quantiles. To estimate these functions, we use the idea of deconvoluting kernels in conjunction with the ordinary kernel methods. We also discuss a new class of function estimators based on local polynomials.

The paper studies estimation of partially linear hazard regression models with varying coefficients for multivariate survival data. A profile pseudopartiallikelihood estimation method is proposed. The estimation of the parameters of the linear part is accomplished via maximization of the profile pseudopartiallikelihood, whereas the varyingcoefficient functions are considered as nuisance parameters that are profiled out of the likelihood. It is shown that the estimators of the parameters are root <i>n</i> consistent and the estimators of the nonparametric coefficient functions achieve optimal convergence rates. Asymptotic normality is obtained for the estimators of the finite parameters and varyingcoefficient functions. Consistent estimators of the asymptotic variances are derived and empirically tested, which facilitate inference for the model. We prove that the varyingcoefficient functions can be estimated as well as if the

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.