arXiv subjects

Runze Li

Publications and source records attributed to Runze Li.

155 records · Page 9Linked to original sources

arXiv2012

Estimation and testing for partially linear single-index models

In partially linear single-index models, we obtain the semiparametrically efficient profile least-squares estimators of regression coefficients. We also employ the smoothly clipped absolute deviation penalty (SCAD) approach to simultaneously select variables and estimate regression coefficients. We show that the resulting SCAD estimators are consistent and possess the oracle property. Subsequently, we demonstrate that a proposed tuning parameter selector, BIC, identifies the true model consistently. Finally, we develop a linear hypothesis test for the parametric coefficients and a goodness-of-fit test for the nonparametric component, respectively. Monte Carlo studies are also presented.

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arXiv2012

Variable selection in linear mixed effects models

This paper is concerned with the selection and estimation of fixed and random effects in linear mixed effects models. We propose a class of nonconcave penalized profile likelihood methods for selecting and estimating important fixed effects. To overcome the difficulty of unknown covariance matrix of random effects, we propose to use a proxy matrix in the penalized profile likelihood. We establish conditions on the choice of the proxy matrix and show that the proposed procedure enjoys the model selection consistency where the number of fixed effects is allowed to grow exponentially with the sample size. We further propose a group variable selection strategy to simultaneously select and estimate important random effects, where the unknown covariance matrix of random effects is replaced with a proxy matrix. We prove that, with the proxy matrix appropriately chosen, the proposed procedure can identify all true random effects with asymptotic probability one, where the dimension of random effects vector is allowed to increase exponentially with the sample size. Monte Carlo simulation studies are conducted to examine the finite-sample performance of the proposed procedures. We further illustrate the proposed procedures via a real data example.

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arXiv2012

Feature Screening via Distance Correlation Learning

This paper is concerned with screening features in ultrahigh dimensional data analysis, which has become increasingly important in diverse scientific fields. We develop a sure independence screening procedure based on the distance correlation (DC-SIS, for short). The DC-SIS can be implemented as easily as the sure independence screening procedure based on the Pearson correlation (SIS, for short) proposed by Fan and Lv (2008). However, the DC-SIS can significantly improve the SIS. Fan and Lv (2008) established the sure screening property for the SIS based on linear models, but the sure screening property is valid for the DC-SIS under more general settings including linear models. Furthermore, the implementation of the DC-SIS does not require model specification (e.g., linear model or generalized linear model) for responses or predictors. This is a very appealing property in ultrahigh dimensional data analysis. Moreover, the DC-SIS can be used directly to screen grouped predictor variables and for multivariate response variables. We establish the sure screening property for the DC-SIS, and conduct simulations to examine its finite sample performance. Numerical comparison indicates that the DC-SIS performs much better than the SIS in various models. We also illustrate the DC-SIS through a real data example.

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arXiv2011

New efficient estimation and variable selection methods for semiparametric varying-coefficient partially linear models

The complexity of semiparametric models poses new challenges to statistical inference and model selection that frequently arise from real applications. In this work, we propose new estimation and variable selection procedures for the semiparametric varying-coefficient partially linear model. We first study quantile regression estimates for the nonparametric varying-coefficient functions and the parametric regression coefficients. To achieve nice efficiency properties, we further develop a semiparametric composite quantile regression procedure. We establish the asymptotic normality of proposed estimators for both the parametric and nonparametric parts and show that the estimators achieve the best convergence rate. Moreover, we show that the proposed method is much more efficient than the least-squares-based method for many non-normal errors and that it only loses a small amount of efficiency for normal errors. In addition, it is shown that the loss in efficiency is at most 11.1% for estimating varying coefficient functions and is no greater than 13.6% for estimating parametric components. To achieve sparsity with high-dimensional covariates, we propose adaptive penalization methods for variable selection in the semiparametric varying-coefficient partially linear model and prove that the methods possess the oracle property. Extensive Monte Carlo simulation studies are conducted to examine the finite-sample performance of the proposed procedures. Finally, we apply the new methods to analyze the plasma beta-carotene level data.

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arXiv2010

Variable selection in measurement error models

Measurement error data or errors-in-variable data have been collected in many studies. Natural criterion functions are often unavailable for general functional measurement error models due to the lack of information on the distribution of the unobservable covariates. Typically, the parameter estimation is via solving estimating equations. In addition, the construction of such estimating equations routinely requires solving integral equations, hence the computation is often much more intensive compared with ordinary regression models. Because of these difficulties, traditional best subset variable selection procedures are not applicable, and in the measurement error model context, variable selection remains an unsolved issue. In this paper, we develop a framework for variable selection in measurement error models via penalized estimating equations. We first propose a class of selection procedures for general parametric measurement error models and for general semi-parametric measurement error models, and study the asymptotic properties of the proposed procedures. Then, under certain regularity conditions and with a properly chosen regularization parameter, we demonstrate that the proposed procedure performs as well as an oracle procedure. We assess the finite sample performance via Monte Carlo simulation studies and illustrate the proposed methodology through the empirical analysis of a familiar data set.

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arXiv2008

One-step sparse estimates in nonconcave penalized likelihood models

Fan and Li propose a family of variable selection methods via penalized likelihood using concave penalty functions. The nonconcave penalized likelihood estimators enjoy the oracle properties, but maximizing the penalized likelihood function is computationally challenging, because the objective function is nondifferentiable and nonconcave. In this article, we propose a new unified algorithm based on the local linear approximation (LLA) for maximizing the penalized likelihood for a broad class of concave penalty functions. Convergence and other theoretical properties of the LLA algorithm are established. A distinguished feature of the LLA algorithm is that at each LLA step, the LLA estimator can naturally adopt a sparse representation. Thus, we suggest using the one-step LLA estimator from the LLA algorithm as the final estimates. Statistically, we show that if the regularization parameter is appropriately chosen, the one-step LLA estimates enjoy the oracle properties with good initial estimators. Computationally, the one-step LLA estimation methods dramatically reduce the computational cost in maximizing the nonconcave penalized likelihood. We conduct some Monte Carlo simulation to assess the finite sample performance of the one-step sparse estimation methods. The results are very encouraging.

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arXiv2008

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.

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arXiv2008

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.

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arXiv2006

Majorization framework for balanced lattice designs

This paper aims to generalize and unify classical criteria for comparisons of balanced lattice designs, including fractional factorial designs, supersaturated designs and uniform designs. We present a general majorization framework for assessing designs, which includes a stringent criterion of majorization via pairwise coincidences and flexible surrogates via convex functions. Classical orthogonality, aberration and uniformity criteria are unified by choosing combinatorial and exponential kernels. A construction method is also sketched out.

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arXiv2006

Statistical Challenges with High Dimensionality: Feature Selection in Knowledge Discovery

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.

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arXiv2005

Variable selection using MM algorithms

Variable selection is fundamental to high-dimensional statistical modeling. Many variable selection techniques may be implemented by maximum penalized likelihood using various penalty functions. Optimizing the penalized likelihood function is often challenging because it may be nondifferentiable and/or nonconcave. This article proposes a new class of algorithms for finding a maximizer of the penalized likelihood for a broad class of penalty functions. These algorithms operate by perturbing the penalty function slightly to render it differentiable, then optimizing this differentiable function using a minorize-maximize (MM) algorithm. MM algorithms are useful extensions of the well-known class of EM algorithms, a fact that allows us to analyze the local and global convergence of the proposed algorithm using some of the techniques employed for EM algorithms. In particular, we prove that when our MM algorithms converge, they must converge to a desirable point; we also discuss conditions under which this convergence may be guaranteed. We exploit the Newton-Raphson-like aspect of these algorithms to propose a sandwich estimator for the standard errors of the estimators. Our method performs well in numerical tests.

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