Difference based Ridge and Liu type Estimators in Semiparametric Regression Models

We consider a difference based ridge regression estimator and a Liu type estimator of the regression parameters in the partial linear semiparametric regression model, y = Xβ + f + ε. Both estimators are analysed and compared in the sense of mean-squared error. We consider the case of independent errors with equal variance and give conditions under which the proposed estimators are superior to the unbiased difference based estimation technique. We extend the results to account for heteroscedasticity and autocovariance in the error terms. Finally, we illustrate the performance of these estimators with an application to the determinants of electricity consumption in Germany.Difference based estimator; Differencing estimator, Differencing matrix, Liu estimator, Liu type estimator, Multicollinearity, Ridge regression estimator, Semiparametric model

Variable selection in semiparametric regression modeling

In this paper, we are concerned with how to select significant variables in semiparametric modeling. Variable selection for semiparametric regression models consists of two components: model selection for nonparametric components and selection of significant variables for the parametric portion. Thus, semiparametric variable selection is much more challenging than parametric variable selection (e.g., linear and generalized linear models) because traditional variable selection procedures including stepwise regression and the best subset selection now require separate model selection for the nonparametric components for each submodel. This leads to a very heavy computational burden. In this paper, we propose a class of variable selection procedures for semiparametric regression models using nonconcave penalized likelihood. We establish the rate of convergence of the resulting estimate. With proper choices of penalty functions and regularization parameters, we show the asymptotic normality of the resulting estimate and further demonstrate that the proposed procedures perform as well as an oracle procedure. A semiparametric generalized likelihood ratio test is proposed to select significant variables in the nonparametric component. We investigate the asymptotic behavior of the proposed test and demonstrate that its limiting null distribution follows a chi-square distribution which is independent of the nuisance parameters. Extensive Monte Carlo simulation studies are conducted to examine the finite sample performance of the proposed variable selection procedures.Comment: Published in at http://dx.doi.org/10.1214/009053607000000604 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

High-dimensional Structured Additive Regression Models: Bayesian Regularisation, Smoothing and Predictive Performance

Data structures in modern applications frequently combine the necessity of flexible regression techniques such as nonlinear and spatial effects with high-dimensional covariate vectors. While estimation of the former is typically achieved by supplementing the likelihood with a suitable smoothness penalty, the latter are usually assigned shrinkage penalties that enforce sparse models. In this paper, we consider a Bayesian unifying perspective, where conditionally Gaussian priors can be assigned to all types of regression effects. Suitable hyperprior assumptions on the variances of the Gaussian distributions then induce the desired smoothness or sparseness properties. As a major advantage, general Markov chain Monte Carlo simulation algorithms can be developed that allow for the joint estimation of smooth and spatial effects and regularised coefficient vectors. Two applications demonstrate the usefulness of the proposed procedure: A geoadditive regression model for data from the Munich rental guide and an additive probit model for the prediction of consumer credit defaults. In both cases, high-dimensional vectors of categorical covariates will be included in the regression models. The predictive ability of the resulting high-dimensional structure additive regression models compared to expert models will be of particular relevance and will be evaluated on cross-validation test data

Semiparametric estimation for a class of time-inhomogenous diffusion processes

Copyright @ 2009 Institute of Statistical Science, Academia SinicaWe develop two likelihood-based approaches to semiparametrically estimate a class of time-inhomogeneous diffusion processes: log penalized splines (P-splines) and the local log-linear method. Positive volatility is naturally embedded and this positivity is not guaranteed in most existing diffusion models. We investigate different smoothing parameter selections. Separate bandwidths are used for drift and volatility estimation. In the log P-splines approach, different smoothness for different time varying coefficients is feasible by assigning different penalty parameters. We also provide theorems for both approaches and report statistical inference results. Finally, we present a case study using the weekly three-month Treasury bill data from 1954 to 2004. We find that the log P-splines approach seems to capture the volatility dip in mid-1960s the best. We also present an application to calculate a financial market risk measure called Value at Risk (VaR) using statistical estimates from log P-splines

Rejoinder: One-step sparse estimates in nonconcave penalized likelihood models

We would like to take this opportunity to thank the discussants for their thoughtful comments and encouragements on our work [arXiv:0808.1012]. The discussants raised a number of issues from theoretical as well as computational perspectives. Our rejoinder will try to provide some insights into these issues and address specific questions asked by the discussants.Comment: Published in at http://dx.doi.org/10.1214/07-AOS0316REJ the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Comparison of Some Suggested Estimators Based on Differencing Technique in the Partial Linear Model Using Simulation

في هذا البحث تم تقديم طرائق مبنية على اساس تقنية الفروق وهي مقدر انحدار الحرف الجاكنايف المعدل المعمم على اساس الفروق (DMJGR) ومقدر انحدار الحرف الجاكنايف المعمم على اساس الفروق (DJGR)  في تقدير معلمات الجزء الخطي من النموذج الخطي الجزئي. أما بالنسبة للجزء غير الخطي الممثل بالدالة اللامعلمية، فقد تم تقديره باستخدام مقدر Nadaraya Watson . تمت مقارنة النموذج الخطي الجزئي باستخدام هذه الطرائق المقترحة مع مقدرات أخرى تعتمد على تقنية الفروق من خلال معيار مقارنة MSE في دراسة محاكاة.In this paper new methods were presented based on technique of differences which is the difference- based modified jackknifed generalized ridge regression estimator(DMJGR) and difference-based generalized  jackknifed ridge regression estimator(DGJR), in estimating the parameters of linear part of the partially linear model. As for the nonlinear part represented by the nonparametric function, it was estimated using Nadaraya Watson smoother. The partially linear model was compared using these proposed methods with other estimators based on differencing technique through the MSE comparison criterion in simulation study