Leming Qu

Wavelet estimation of partially linear models

A wavelet approachis presented for estimating a partially linear model (PLM). We /nd an estimator of the PLM by minimizing the square of the l2 norm of the residual vector while penalizing the l1 norm of the wavelet coe2cients of the nonparametric component. This approach, an extension of the wavelet approach for nonparametric regression problems, avoids the restrictive smoothness requirements for the nonparametric function of the traditional smoothing approaches for PLM, such as smoothing spline, kernel and piecewise polynomial methods. To solve the optimization problem, an e2cient descent algorith m withan exact line searchis presented. Simulation results are given to demonstrate e6ectiveness of our method. c 2003 Elsevier B.V. All rights reserved.

Copula density estimation by total variation penalized likelihood with linear equality constraints

A copula density is the joint probability density function (PDF) of a random vector with uniform marginals. An approach to bivariate copula density estimation is introduced that is based on maximum penalized likelihood estimation (MPLE) with a total variation (TV) penalty term. The marginal unity and symmetry constraints for copula density are enforced by linear equality constraints. The TV-MPLE subject to linear equality constraints is solved by an augmented Lagrangian and operator-splitting algorithm. It offers an order of magnitude improvement in computational efficiency over another TV-MPLE method without constraints solved by the log-barrier method for the second order cone program. A data-driven selection of the regularization parameter is through K-fold cross-validation (CV). Simulation and real data application show the effectiveness of the proposed approach. The MATLAB code implementing the methodology is available online.

Bayesian Wavelet Estimation of Partially Linear Models

A Bayesian wavelet approach is presented for estimating a partially linear model (PLM). A PLM consists of a linear part and a nonparametric component. The nonparametric component is represented with a wavelet series where the wavelet coefficients have assumed prior distributions. The prior for each coefficient consists of a mixture of a normal distribution and a point mass at 0. The linear parameters are assumed to have a normal prior. The hyperparameters are estimated by the marginal maximum likelihood estimator using the direct maximization. The model selection and model averaging methods give different estimates of the model parameters. MCMC computation is used for the estimation of the linear coefficients by model averaging method. Simulated examples illustrate the performance of the proposed estimators.

Copula Density Estimation by Total Variation Penalized Likelihood

Copulas are full measures of dependence among random variables. They are increasingly popular among academics and practitioners in financial econometrics for modeling comovements between markets, risk factors, and other relevant variables. A copulas hidden dependence structure that couples a joint distribution with its marginals makes a parametric copula non-trivial. An approach to bivariate copula density estimation is introduced that is based on a penalized likelihood with a total variation penalty term. Adaptive choice of the amount of regularization is based on approximate Bayesian Information Criterion (BIC) type scores. Performance are evaluated through the Monte Carlo simulation.

Wavelet Reconstruction of Nonuniformly Sampled Signals

For the reconstruction of a nonuniformly sampled signal based on its noisy observations, we propose a level dependent l1 penalized wavelet reconstruction method. The LARS/Lasso algorithm is applied to solve the Lasso problem. The data adaptive choice of the regularization parameters is based on the AIC and the degrees of freedom is estimated by the number of nonzero elements in the Lasso solution. Simulation results conducted on some commonly used 1_D test signals illustrate that the proposed method possesses good empirical properties.

Wavelet deconvolution in a periodic setting using cross-validation

The wavelet deconvolution method WaveD using band-limited wavelets offers both theoretical and computational advantages over traditional compactly supported wavelets. The translation-invariant WaveD with a fast algorithm improves further. The twofold cross-validation method for choosing the threshold parameter and the finest resolution level in WaveD is introduced. The algorithms performance is compared with the fixed constant tuning and the default tuning in WaveD.