Lixin Song

Linear B-spline copulas with applications to nonparametric estimation of copulas

In this paper, we propose a method for constructing a new class of copulas. They are called linear B-spline copulas which are a good approximation of a given complicated copula by using finite numbers of values of this copula without the loss of some essential properties. Moreover, rigorous analysis shows that the empirical linear B-spline copulas are more effective than empirical copulas to estimate perfectly dependent copulas. For the cases of nonperfectly dependent copulas, simulations show that the empirical linear B-spline copulas also improve the empirical copulas to estimate the underlying copula structure. Furthermore, we compare the performance of parametric estimation of copulas based on the empirical copulas with that based on the empirical linear B-spline copulas by simulations. In most of the cases, the latter are better than the former.

Quadratic Approximation via the SCAD Penalty with a Diverging Number of Parameters

The high-dimensional data arises in diverse fields of sciences, engineering and humanities. Variable selection plays an important role in dealing with high dimensional statistical modelling. In this article, we study the variable selection of quadratic approximation via the smoothly clipped absolute deviation (SCAD) penalty with a diverging number of parameters. We provide a unified method to select variables and estimate parameters for various of high dimensional models. Under appropriate conditions and with a proper regularization parameter, we show that the estimator has consistency and sparsity, and the estimators of nonzero coefficients enjoy the asymptotic normality as they would have if the zero coefficients were known in advance. In addition, under some mild conditions, we can obtain the global solution of the penalized objective function with the SCAD penalty. Numerical studies and a real data analysis are carried out to confirm the performance of the proposed method.

Variable selection in the high-dimensional continuous generalized linear model with current status data

In survival studies, current status data are frequently encountered when some individuals in a study are not successively observed. This paper considers the problem of simultaneous variable selection and parameter estimation in the high-dimensional continuous generalized linear model with current status data. We apply the penalized likelihood procedure with the smoothly clipped absolute deviation penalty to select significant variables and estimate the corresponding regression coefficients. With a proper choice of tuning parameters, the resulting estimator is shown to be a root n/pn-consistent estimator under some mild conditions. In addition, we show that the resulting estimator has the same asymptotic distribution as the estimator obtained when the true model is known. The finite sample behavior of the proposed estimator is evaluated through simulation studies and a real example.

SCAD-Penalized Least Absolute Deviation Regression in High-Dimensional Models

When outliers and/or heavy-tailed errors exist in linear models, the least absolute deviation (LAD) regression is a robust alternative to the ordinary least squares regression. Existing variable-selection methods in linear models based on LAD regression either only consider the finite number of predictors or lack the oracle property associated with the estimator. In this article, we focus on the variable selection via LAD regression with a diverging number of parameters. The rate of convergence of the LAD estimator with the smoothly clipped absolute deviation (SCAD) penalty function is established. Furthermore, we demonstrate that, under certain regularity conditions, the penalized estimator with a properly selected tuning parameter enjoys the oracle property. In addition, the rank correlation screening method originally proposed by Li et al. (2011) is applied to deal with ultrahigh dimensional data. Simulation studies are conducted for revealing the finite sample performance of the estimator. We further illustrate the proposed methodology by a real example.

Bridge Estimators in the Partially Linear Model with High Dimensionality

This article studies variable selection and parameter estimation in the partially linear model when the number of covariates in the linear part increases to infinity. Using the bridge penalty method, we succeed in selecting the important covariates of the linear part. Under regularity conditions, we have shown that the bridge penalized estimator of the parametric part enjoys the oracle property. We also obtain the convergence rate of the estimator of the nonparametric part. Simulation studies show that the bridge estimator performs as well as the oracle estimator for the partially linear model. An application is analyzed to illustrate the bridge procedure.

Combined-penalized likelihood estimations with a diverging number of parameters

In the economics and biological gene expression study area where a large number of variables will be involved, even when the predictors are independent, as long as the dimension is high, the maximum sample correlation can be large. Variable selection is a fundamental method to deal with such models. The ridge regression performs well when the predictors are highly correlated and some nonconcave penalized thresholding estimators enjoy the nice oracle property. In order to provide a satisfactory solution to the collinearity problem, in this paper we report the combined-penalization (CP) mixed by the nonconcave penalty and ridge, with a diverging number of parameters. It is observed that the CP estimator with a diverging number of parameters can correctly select covariates with nonzero coefficients and can estimate parameters simultaneously in the presence of multicollinearity. Simulation studies and a real data example demonstrate the well performance of the proposed method.

Asymptotic Behavior of Random Time Ruin Probability Under Heavy-Tailed Claim Sizes and Dependence Structure

This article deals with the renewal risk model, in which there exists some asymptotic dependence relation between claim sizes and the inter-arrival times, and claim sizes are subexponential. Under this setting, we investigate the tail behaviour of random time ruin probability as the initial risk reserve x tends to infinity. We obtain the similar asymptotic formula as the previous results.

Adaptive Lasso Variable Selection for the Accelerated Failure Models

This article considers the adaptive lasso procedure for the accelerated failure time model with multiple covariates based on weighted least squares method, which uses Kaplan-Meier weights to account for censoring. The adaptive lasso method can complete the variable selection and model estimation simultaneously. Under some mild conditions, the estimator is shown to have sparse and oracle properties. We use Bayesian Information Criterion (BIC) for tuning parameter selection, and a bootstrap variance approach for standard error. Simulation studies and two real data examples are carried out to investigate the performance of the proposed method.

A quicker continued fraction approximation of the gamma function related to the Windschitl's formula

In this paper, based on the Windschitl’s formula, a new continued fraction approximation and some inequalities for the gamma function are established. Finally, for demonstrating the superiority of our new series over the classical ones, some numerical computations are given.

Penalized Empirical Likelihood via Bridge Estimator in Cox's Proportional Hazard Model

We propose the penalized empirical likelihood method via bridge estimator in Coxs proportional hazard model for parameter estimation and variable selection. Under reasonable conditions, we show that penalized empirical likelihood in Coxs proportional hazard model has oracle property. A penalized empirical likelihood ratio for the vector of regression coefficients is defined and its limiting distribution is a chi-square distributions. The advantage of penalized empirical likelihood as a nonparametric likelihood approach is illustrated in testing hypothesis and constructing confidence sets. The method is illustrated by extensive simulation studies and a real example.