A note on smoothing parameter selection for penalized spline smoothing

In nonparametric regression the smoothing parameter can be selected by minimizing a Mean Squared Error (MSE) based criterion. For spline smoothing one can also rewrite the smooth estimation as a Linear Mixed Model where the smoothing parameter appears as the a priori variance of spline basis coefficients. This allows to employ Maximum Likelihood (ML) theory to estimate the smoothing parameter as variance component. In this paper the relation between the two approaches is illuminated for penalized spline smoothing (P-spline) as suggested in Eilers and Marx Statist. Sci. 11(2) (1996) 89. Theoretical and empirical arguments are given showing that the ML approach is biased towards undersmoothing, i.e. it chooses a too complex model compared to the MSE. The result is in line with classical spline smoothing, even though the asymptotic arguments are different. This is because in P-spline smoothing a finite dimensional basis is employed while in classical spline smoothing the basis grows with the sample size.     

Generalized varying-coefficient single-index model

In this paper, the generalized varying-coefficient single-index model is discussed based on penalized likelihood. All the unknown functions are fitted by penalized spline. The estimates of the unknown parameters and the unknown coefficient functions are obtained and the estimation approach is rapid and computationally stable. Under some mild conditions, the consistency and the asymptotic normality of these resulting estimators are given. Two simulation studies are carried out to illustrate the performance of the estimates. An application of the model to the Hong Kong environmental data further demonstrates the potential of the proposed modelling procedures.     

Testing the linearity of negative binomial regression models

The negative binomial (NB) is frequently used to model overdispersed Poisson count data. To study the effect of a continuous covariate of interest in an NB model, a flexible procedure is used to model the covariate effect by fixed-knot cubic basis-splines or B-splines with a second-order difference penalty on the adjacent B-spline coefficients to avoid undersmoothing. A penalized likelihood is used to estimate parameters of the model. A penalized likelihood ratio test statistic is constructed for the null hypothesis of the linearity of the continuous covariate effect. When the number of knots is fixed, its limiting null distribution is the distribution of a linear combination of independent chi-squared random variables, each with one degree of freedom. The smoothing parameter value is determined by setting a specified value equal to the asymptotic expectation of the test statistic under the null hypothesis. The power performance of the proposed test is studied with simulation experiments.     

Penalized Spline Estimation for Varying-Coefficient Models

Varying-coefficient models are useful extensions of classical linear models. They arise from multivariate nonparametric regression, nonlinear time series modeling and forecasting, longitudinal data analysis, and others. This article proposes the penalized spline estimation for the varying-coefficient models. Assuming a fixed but potentially large number of knots, the penalized spline estimators are shown to be strong consistency and asymptotic normality. A systematic optimization algorithm for the selection of multiple smoothing parameters is developed. One of the advantages of the penalized spline estimation is that it can accommodate varying degrees of smoothness among coefficient functions due to multiple smoothing parameters being used. Some simulation studies are presented to illustrate the proposed methods.     

Using -splines to test the linearity of partially linear models

The nonparametric component in a partially linear model is estimated by a linear combination of fixed-knot cubic B-splines with a second-order difference penalty on the adjacent B-spline coefficients. The resulting penalized least-squares estimator is used to construct two Wald-type spline-based test statistics for the null hypothesis of the linearity of the nonparametric function. When the number of knots is fixed, the first test statistic asymptotically has the distribution of a linear combination of independent chi-squared random variables, each with one degree of freedom, under the null hypothesis. The smoothing parameter is determined by specifying a value for the asymptotically expected value of the test statistic under the null hypothesis. When the number of knots is fixed and under the null hypothesis, the second test statistic asymptotically has a chi-squared distribution with K=q+2 degrees of freedom, where q is the number of knots used for estimation. The power performances of the two proposed tests are investigated via simulation experiments, and the practicality of the proposed methodology is illustrated using a real-life data set.     

Constrained penalized splines

The penalized spline is a popular method for function estimation when the assumption of “smoothness” is valid. In this paper, methods for estimation and inference are proposed using penalized splines under additional constraints of shape, such as monotonicity or convexity. The constrained penalized spline estimator is shown to have the same convergence rates as the corresponding unconstrained penalized spline, although in practice the squared error loss is typically smaller for the constrained versions. The penalty parameter may be chosen with generalized cross‐validation, which also provides a method for determining if the shape restrictions hold. The method is not a formal hypothesis test, but is shown to have nice large‐sample properties, and simulations show that it compares well with existing tests for monotonicity. Extensions to the partial linear model, the generalized regression model, and the varying coefficient model are given, and examples demonstrate the utility of the methods. The Canadian Journal of Statistics 40: 190–206; 2012 © 2012 Statistical Society of Canada     

Efficient Penalized Estimation for Linear Regression Model

This paper develops new penalized estimation for linear regression model. We prove that the new method, which is referred to as efficient penalized estimation, is selection consistent, and more asymptotically efficient than the original one. Besides, we construct a new selector called efficient BIC Selector to tune the regularization parameter in the new estimation, which is shown to be consistent. Our simulation results suggest that the new method may bring significant improvement relative to the original penalized estimation. In addition, we employ a real data set to illustrate the application of the efficient penalized estimation.     

Simultaneous sparse model selection and coefficient estimation for heavy-tailed autoregressive processes

We propose a sparse coefficient estimation and automated model selection procedure for autoregressive processes with heavy-tailed innovations based on penalized conditional maximum likelihood. Under mild moment conditions on the innovation processes, the penalized conditional maximum likelihood estimator satisfies a strong consistency, O_P(N^?1/2) consistency, and the oracle properties, where N is the sample size. We have the freedom in choosing penalty functions based on the weak conditions on them. Two penalty functions, least absolute shrinkage and selection operator and smoothly clipped average deviation, are compared. The proposed method provides a distribution-based penalized inference to AR models, which is especially useful when the other estimation methods fail or under perform for AR processes with heavy-tailed innovations [Feigin, Resnick. Pitfalls of fitting autoregressive models for heavy-tailed time series. Extremes. 1999;1:391–422]. A simulation study confirms our theoretical results. At the end, we apply our method to a historical price data of the US Industrial Production Index for consumer goods, and obtain very promising results.     

Penalized least-squares estimation for regression coefficients in high-dimensional partially linear models

We consider a partially linear model with diverging number of groups of parameters in the parametric component. The variable selection and estimation of regression coefficients are achieved simultaneously by using the suitable penalty function for covariates in the parametric component. An MM-type algorithm for estimating parameters without inverting a high-dimensional matrix is proposed. The consistency and sparsity of penalized least-squares estimators of regression coefficients are discussed under the setting of some nonzero regression coefficients with very small values. It is found that the root p_n/n-consistency and sparsity of the penalized least-squares estimators of regression coefficients cannot be given consideration simultaneously when the number of nonzero regression coefficients with very small values is unknown, where p_n and n, respectively, denote the number of regression coefficients and sample size. The finite sample behaviors of penalized least-squares estimators of regression coefficients and the performance of the proposed algorithm are studied by simulation studies and a real data example.     

Small area estimation of the mean using non-parametric M-quantile regression: a comparison when a linear mixed model does not hold

The demand for reliable statistics in subpopulations, when only reduced sample sizes are available, has promoted the development of small area estimation methods. In particular, an approach that is now widely used is based on the seminal work by Battese et al. [An error-components model for prediction of county crop areas using survey and satellite data, J. Am. Statist. Assoc. 83 (1988), pp. 28–36] that uses linear mixed models (MM). We investigate alternatives when a linear MM does not hold because, on one side, linearity may not be assumed and/or, on the other, normality of the random effects may not be assumed. In particular, Opsomer et al. [Nonparametric small area estimation using penalized spline regression, J. R. Statist. Soc. Ser. B 70 (2008), pp. 265–283] propose an estimator that extends the linear MM approach to the case in which a linear relationship may not be assumed using penalized splines regression. From a very different perspective, Chambers and Tzavidis [M-quantile models for small area estimation, Biometrika 93 (2006), pp. 255–268] have recently proposed an approach for small-area estimation that is based on M-quantile (MQ) regression. This allows for models robust to outliers and to distributional assumptions on the errors and the area effects. However, when the functional form of the relationship between the qth MQ and the covariates is not linear, it can lead to biased estimates of the small area parameters. Pratesi et al. [Semiparametric M-quantile regression for estimating the proportion of acidic lakes in 8-digit HUCs of the Northeastern US, Environmetrics 19(7) (2008), pp. 687–701] apply an extended version of this approach for the estimation of the small area distribution function using a non-parametric specification of the conditional MQ of the response variable given the covariates [M. Pratesi, M.G. Ranalli, and N. Salvati, Nonparametric m-quantile regression using penalized splines, J. Nonparametric Stat. 21 (2009), pp. 287–304]. We will derive the small area estimator of the mean under this model, together with its mean-squared error estimator and compare its performance to the other estimators via simulations on both real and simulated data.     

ESTIMATING EQUATIONS AND NON-LINEAR FUNCTIONAL RELATIONSHIPS

A nonlinear functional relationship is defined as an R-dimensional manifold in P-dimensional space. The formulation of the model may be explicitly in terms of R-dimensional vectors of incidental parameters or implicitly by a (P-R)-dimensional vector function of constraints. The objective is to estimate and make inference about a K-vector of parameters θ which defines the manifold. Each observed P-vector has its expectation lying on the manifold, and the error vector has a variance matrix defined in terms of a further vector of parameters The theory of estimating equations in the presence of incidental parameters is extended and applied to the explicit formulation, to give equations suitable for estimating θ given knowledge of only the first two moments. The method has a geometrical interpretation. Estimating equations for are chosen to be those which would be optimal if the normality assumption were true. First order corrections to the biases of these estimates are included. An example where the manifold is a circle centred on the origin is used to illustrate the theory. Further examples incorporate more general features, including the estimation of two variance parameters and estimation in higher dimensions.     

Improvements in the small sample efficiency of the minimum S-divergence estimators under discrete models

This paper considers the problem of inliers and empty cells and the resulting issue of relative inefficiency in estimation under pure samples from a discrete population when the sample size is small. Many minimum divergence estimators in the S-divergence family, although possessing very strong outlier stability properties, often have very poor small sample efficiency in the presence of inliers and some are not even defined in the presence of a single empty cell; this limits the practical applicability of these estimators, in spite of their otherwise sound robustness properties and high asymptotic efficiency. Here, we study a penalized version of the S-divergences such that the resulting minimum divergence estimators are free from these issues, without altering their robustness properties and asymptotic efficiencies. We present a general proof for the asymptotic properties of these minimum penalized S-divergence estimators. This provides a significant addition to the literature, as the asymptotics of penalized divergences which are not finitely defined are currently unavailable in the literature. The small sample advantages of the minimum penalized S-divergence estimators are examined through an extensive simulation study and some empirical suggestions regarding the choice of the relevant underlying tuning parameters are also provided.     

Maximum penalized likelihood estimation of mixed proportional hazard models

Unobservable individual effects in models of duration will cause estimation bias that include the structural parameters as well as the duration dependence. The maximum penalized likelihood estimator is examined as an estimator for the survivor model with heterogeneity. Proofs of the existence and uniqueness of the maximum penalized likelihood estimator in duration model with general forms of unobserved heterogeneity are provided. Some small sample evidence on the behavior of the maximum penalized likelihood estimator is given. The maximum penalized likelihood estimator is shown to be computationally feasible and to provide reasonable estimates in most cases.     

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/p_n-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.     

On approximate likelihood inference in a poisson mixed model

A two-step estimation approach is proposed for the fixed-effect parameters, random effects and their variance σ² of a Poisson mixed model. In the first step, it is proposed to construct a small σ²-based approximate likelihood function of the data and utilize this function to estimate the fixed-effect parameters and σ². In the second step, the random effects are estimated by minimizing their posterior mean squared error. Methods of Waclawiw and Liang (1993) based on so-called Stein-type estimating functions and of Breslow and Clayton (1993) based on penalized quasilikelihood are compared with the proposed likelihood method. The results of a simulation study on the performance of all three approaches are reported.     

A simple approach for varying-coefficient model selection

In varying-coefficient models, an important question is to determine whether some of the varying coefficients are actually invariant coefficients. This article proposes a penalized likelihood method in the framework of the smoothing spline ANOVA models, with a penalty designed toward the goal of automatically distinguishing varying coefficients and those which are not varying. Unlike the stepwise procedure, the method simultaneously quantifies and estimates the coefficients. An efficient algorithm is given and ways of choosing the smoothing parameters are discussed. Simulation results and an analysis on the Boston housing data illustrate the usefulness of the method. The proposed approach is further extended to longitudinal data analysis.     

Robust sparse regression and tuning parameter selection via the efficient bootstrap information criteria

There is currently much discussion about lasso-type regularized regression which is a useful tool for simultaneous estimation and variable selection. Although the lasso-type regularization has several advantages in regression modelling, owing to its sparsity, it suffers from outliers because of using penalized least-squares methods. To overcome this issue, we propose a robust lasso-type estimation procedure that uses the robust criteria as the loss function, imposing L₁-type penalty called the elastic net. We also introduce to use the efficient bootstrap information criteria for choosing optimal regularization parameters and a constant in outlier detection. Simulation studies and real data analysis are given to examine the efficiency of the proposed robust sparse regression modelling. We observe that our modelling strategy performs well in the presence of outliers.     

Estimation on Varying-Coefficient Partially Linear Model With Different Smoothing Variables

The authors give the estimation on the varying-coefficient partially linear regression model with different smoothing variables. The efficient estimators of the intercept function and the coefficient functions are obtained by a one-step back-fitting technique based on their initial estimators given by local linear technique and the averaged method. Furthermore, their asymptotic normalities are given. Some simulation studies are used to illustrate the performances of the estimation.     

Two‐stage model selection procedures in partially linear regression

The authors propose a two‐stage estimation procedure for the partially linear model Y = f_o(T) + X'β_o + ψ. They show how to estimate consistently the location of the nonzero components of β_o. Their approach turns out to be compatible with minimax adaptive estimation of f_o over Besov balls in the case of penalized least squares. Their proofs are based on a new type of oracle inequality.     

N-stage splitting for maximum penalized likelihood estimation

An n-stage splitting algorithm for the solution of maximum penalized likelihood estimation (MPLE) problems is compared to the one-step-late (OSL) algorithm. General conditions under which the asymptotic rate of convergence of this splitting algorithm. exceeds that of the OSL algorithm are given. A one-dimensional positive data example, illustrates the comparison of the rates of convergence of these two algorithms.