Technological innovations have revolutionized the process of scientific research and knowledge discovery. The availability of massive data and challenges from frontiers of research and development have reshaped statistical thinking, data analysis and theoretical studies. The challenges of high-dimensionality arise in diverse fields of sciences and the humanities, ranging from computational biology and health studies to financial engineering and risk management. In all of these fields, variable selection and feature extraction are crucial for knowledge discovery. We first give a comprehensive overview of statistical challenges with high dimensionality in these diverse disciplines. We then approach the problem of variable selection and feature extraction using a unified framework: penalized likelihood methods. Issues relevant to the choice of penalty functions are addressed. We demonstrate that for a host of statistical problems, as long as the dimensionality is not excessively large, we can estimate the model parameters as well as if the best model is known in advance. The persistence property in risk minimization is also addressed. The applicability of such a theory and method to diverse statistical problems is demonstrated. Other related problems with high-dimensionality are also discussed.

There are few techniques available for testing whether or not a family of parametric times series models fits a set of data reasonably well without serious restrictions on the forms of alternative models. In this paper, we consider generalised likelihood ratio tests of whether or not the spectral density function of a stationary time series admits certain parametric forms. We propose a bias correction method for the generalised likelihood ratio test of Fan et al. (2001). In particular, our methods can be applied to test whether or not a residual series is white noise. Sampling properties of the proposed tests are established. A bootstrap approach is proposed for estimating the null distribution of the test statistics. Simulation studies investigate the accuracy of the proposed bootstrap estimate and compare the power of the various ways of constructing the generalised likelihood ratio tests as well as some classic methods like

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.

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

Value at Risk (VaR) is a fundamental tool for managing market risks. It measures the worst loss to be expected of a portfolio over a given time horizon under normal market conditions at a given confidence level. Calculation of VaR frequently involves estimating the volatility of return processes and quantiles of standardized returns. In this paper, several semiparametric techniques are introduced to estimate the volatilities of the market prices of a portfolio. In addition, both parametric and nonparametric techniques are proposed to estimate the quantiles of standardized return processes. The newly proposed techniques also have the flexibility to adapt automatically to the changes in the dynamics of market prices over time. Their statistical efficiencies are studied both theoretically and empirically. The combination of newly proposed techniques for estimating volatility and standardized quantiles yields several new