Detection of multiple undocumented change-points using adaptive Lasso

The problem of detecting multiple undocumented change-points in a historical temperature sequence with simple linear trend is formulated by a linear model. We apply adaptive least absolute shrinkage and selection operator (Lasso) to estimate the number and locations of change-points. Model selection criteria are used to choose the Lasso smoothing parameter. As adaptive Lasso may overestimate the number of change-points, we perform post-selection on change-points detected by adaptive Lasso using multivariate t simultaneous confidence intervals. Our method is demonstrated on the annual temperature data (year: 1902–2000) from Tuscaloosa, Alabama.     

WLAD-LASSO method for robust estimation and variable selection in partially linear models

This paper focuses on robust estimation and variable selection for partially linear models. We combine the weighted least absolute deviation (WLAD) regression with the adaptive least absolute shrinkage and selection operator (LASSO) to achieve simultaneous robust estimation and variable selection for partially linear models. Compared with the LAD-LASSO method, the WLAD-LASSO method will resist to the heavy-tailed errors and outliers in the parametric components. In addition, we estimate the unknown smooth function by a robust local linear regression. Under some regular conditions, the theoretical properties of the proposed estimators are established. We further examine finite-sample performance of the proposed procedure by simulation studies and a real data example.     

The Risk of James–Stein and Lasso Shrinkage

This article compares the mean-squared error (or ?₂ risk) of ordinary least squares (OLS), James–Stein, and least absolute shrinkage and selection operator (Lasso) shrinkage estimators in simple linear regression where the number of regressors is smaller than the sample size. We compare and contrast the known risk bounds for these estimators, which shows that neither James–Stein nor Lasso uniformly dominates the other. We investigate the finite sample risk using a simple simulation experiment. We find that the risk of Lasso estimation is particularly sensitive to coefficient parameterization, and for a significant portion of the parameter space Lasso has higher mean-squared error than OLS. This investigation suggests that there are potential pitfalls arising with Lasso estimation, and simulation studies need to be more attentive to careful exploration of the parameter space.     

Outlier detection and robust variable selection via the penalized weighted LAD-LASSO method

This paper studies the outlier detection and robust variable selection problem in the linear regression model. The penalized weighted least absolute deviation (PWLAD) regression estimation method and the adaptive least absolute shrinkage and selection operator (LASSO) are combined to simultaneously achieve outlier detection, and robust variable selection. An iterative algorithm is proposed to solve the proposed optimization problem. Monte Carlo studies are evaluated the finite-sample performance of the proposed methods. The results indicate that the finite sample performance of the proposed methods performs better than that of the existing methods when there are leverage points or outliers in the response variable or explanatory variables. Finally, we apply the proposed methodology to analyze two real datasets.     

LAD-Lasso variable selection for doubly censored median regression models

ABSTRACT

A variable selection procedure based on least absolute deviation (LAD) estimation and adaptive lasso (LAD-Lasso for short) is proposed for median regression models with doubly censored data. The proposed procedure can select significant variables and estimate the parameters simultaneously, and the resulting estimators enjoy the oracle property. Simulation results show that the proposed method works well.     

Lassoing the HAR Model: A Model Selection Perspective on Realized Volatility Dynamics

Realized volatility computed from high-frequency data is an important measure for many applications in finance, and its dynamics have been widely investigated. Recent notable advances that perform well include the heterogeneous autoregressive (HAR) model which can approximate long memory, is very parsimonious, is easy to estimate, and features good out-of-sample performance. We prove that the least absolute shrinkage and selection operator (Lasso) recovers the lags structure of the HAR model asymptotically if it is the true model, and we present Monte Carlo evidence in finite samples. The HAR model's lags structure is not fully in agreement with the one found using the Lasso on real data. Moreover, we provide empirical evidence that there are two clear breaks in structure for most of the assets we consider. These results bring into question the appropriateness of the HAR model for realized volatility. Finally, in an out-of-sample analysis, we show equal performance of the HAR model and the Lasso approach.     

Regularisation Parameter Selection Via Bootstrapping

Penalised likelihood methods, such as the least absolute shrinkage and selection operator (Lasso) and the smoothly clipped absolute deviation penalty, have become widely used for variable selection in recent years. These methods impose penalties on regression coefficients to shrink a subset of them towards zero to achieve parameter estimation and model selection simultaneously. The amount of shrinkage is controlled by the regularisation parameter. Popular approaches for choosing the regularisation parameter include cross‐validation, various information criteria and bootstrapping methods that are based on mean square error. In this paper, a new data‐driven method for choosing the regularisation parameter is proposed and the consistency of the method is established. It holds not only for the usual fixed‐dimensional case but also for the divergent setting. Simulation results show that the new method outperforms other popular approaches. An application of the proposed method to motif discovery in gene expression analysis is included in this paper.     

Lassoing the Determinants of Retirement

This article uses Danish register data to explain the retirement decision of workers in 1990 and 1998. Many variables might be conjectured to influence this decision such as demographic, socioeconomic, financial, and health related variables as well as all the same factors for the spouse in case the individual is married. In total, we have access to 399 individual specific variables that all could potentially impact the retirement decision. We use variants of the least absolute shrinkage and selection operator (Lasso) and the adaptive Lasso applied to logistic regression in order to uncover determinants of the retirement decision. To the best of our knowledge, this is the first application of these estimators in microeconometrics to a problem of this type and scale. Furthermore, we investigate whether the factors influencing the retirement decision are stable over time, gender, and marital status. It is found that this is the case for core variables such as age, income, wealth, and general health. We also point out the most important differences between these groups and explain why these might be present.     

Application of shrinkage estimation in linear regression models with autoregressive errors

In this paper, we consider the shrinkage and penalty estimation procedures in the linear regression model with autoregressive errors of order p when it is conjectured that some of the regression parameters are inactive. We develop the statistical properties of the shrinkage estimation method including asymptotic distributional biases and risks. We show that the shrinkage estimators have a significantly higher relative efficiency than the classical estimator. Furthermore, we consider the two penalty estimators: least absolute shrinkage and selection operator (LASSO) and adaptive LASSO estimators, and numerically compare their relative performance with that of the shrinkage estimators. A Monte Carlo simulation experiment is conducted for different combinations of inactive predictors and the performance of each estimator is evaluated in terms of the simulated mean-squared error. This study shows that the shrinkage estimators are comparable to the penalty estimators when the number of inactive predictors in the model is relatively large. The shrinkage and penalty methods are applied to a real data set to illustrate the usefulness of the procedures in practice.     

Oracle model selection for correlated data via residuals

This paper concerns model selection for autoregressive time series when the observations are contaminated with trend. We propose an adaptive least absolute shrinkage and selection operator (LASSO) type model selection method, in which the trend is estimated by B-splines, the detrended residuals are calculated, and then the residuals are used as if they were observations to optimize an adaptive LASSO type objective function. The oracle properties of such an adaptive LASSO model selection procedure are established; that is, the proposed method can identify the true model with probability approaching one as the sample size increases, and the asymptotic properties of estimators are not affected by the replacement of observations with detrended residuals. The intensive simulation studies of several constrained and unconstrained autoregressive models also confirm the theoretical results. The method is illustrated by two time series data sets, the annual U.S. tobacco production and annual tree ring width measurements.     

Shrinkage and penalized estimators in weighted least absolute deviations regression models

In this paper, we consider the estimation problem of the weighted least absolute deviation (WLAD) regression parameter vector when there are some outliers or heavy-tailed errors in the response and the leverage points in the predictors. We propose the pretest and James–Stein shrinkage WLAD estimators when some of the parameters may be subject to certain restrictions. We derive the asymptotic risk of the pretest and shrinkage WLAD estimators and show that if the shrinkage dimension exceeds two, the asymptotic risk of the shrinkage WLAD estimator is strictly less than the unrestricted WLAD estimator. On the other hand, the risk of the pretest WLAD estimator depends on the validity of the restrictions on the parameters. Furthermore, we study the WLAD absolute shrinkage and selection operator (WLAD-LASSO) and compare its relative performance with the pretest and shrinkage WLAD estimators. A simulation study is conducted to evaluate the performance of the proposed estimators relative to that of the unrestricted WLAD estimator. A real-life data example using body fat study is used to illustrate the performance of the suggested estimators.     

New Robust Variable Selection Methods for Linear Regression Models

Motivated by an entropy inequality, we propose for the first time a penalized profile likelihood method for simultaneously selecting significant variables and estimating unknown coefficients in multiple linear regression models in this article. The new method is robust to outliers or errors with heavy tails and works well even for error with infinite variance. Our proposed approach outperforms the adaptive lasso in both theory and practice. It is observed from the simulation studies that (i) the new approach possesses higher probability of correctly selecting the exact model than the least absolute deviation lasso and the adaptively penalized composite quantile regression approach and (ii) exact model selection via our proposed approach is robust regardless of the error distribution. An application to a real dataset is also provided.     

Bayesian adaptive lasso with variational Bayes for variable selection in high-dimensional generalized linear mixed models

This article describes a full Bayesian treatment for simultaneous fixed-effect selection and parameter estimation in high-dimensional generalized linear mixed models. The approach consists of using a Bayesian adaptive Lasso penalty for signal-level adaptive shrinkage and a fast Variational Bayes scheme for estimating the posterior mode of the coefficients. The proposed approach offers several advantages over the existing methods, for example, the adaptive shrinkage parameters are automatically incorporated, no Laplace approximation step is required to integrate out the random effects. The performance of our approach is illustrated on several simulated and real data examples. The algorithm is implemented in the R package glmmvb and is made available online.     

The Adaptive Gril Estimator with a Diverging Number of Parameters

We consider the problem of variables selection and estimation in linear regression model in situations where the number of parameters diverges with the sample size. We propose the adaptive Generalized Ridge-Lasso (\mboxAdaGril) which is an extension of the the adaptive Elastic Net. AdaGril incorporates information redundancy among correlated variables for model selection and estimation. It combines the strengths of the quadratic regularization and the adaptively weighted Lasso shrinkage. In this article, we highlight the grouped selection property for AdaCnet method (one type of AdaGril) in the equal correlation case. Under weak conditions, we establish the oracle property of AdaGril which ensures the optimal large performance when the dimension is high. Consequently, it achieves both goals of handling the problem of collinearity in high dimension and enjoys the oracle property. Moreover, we show that AdaGril estimator achieves a Sparsity Inequality, i.e., a bound in terms of the number of non-zero components of the “true” regression coefficient. This bound is obtained under a similar weak Restricted Eigenvalue (RE) condition used for Lasso. Simulations studies show that some particular cases of AdaGril outperform its competitors.     

Shrinkage averaging estimation

We discuss the impact of tuning parameter selection uncertainty in the context of shrinkage estimation and propose a methodology to account for problems arising from this issue: Transferring established concepts from model averaging to shrinkage estimation yields the concept of shrinkage averaging estimation (SAE) which reflects the idea of using weighted combinations of shrinkage estimators with different tuning parameters to improve overall stability, predictive performance and standard errors of shrinkage estimators. Two distinct approaches for an appropriate weight choice, both of which are inspired by concepts from the recent literature of model averaging, are presented: The first approach relates to an optimal weight choice with regard to the predictive performance of the final weighted estimator and its implementation can be realized via quadratic programming. The second approach has a fairly different motivation and considers the construction of weights via a resampling experiment. Focusing on Ridge, Lasso and Random Lasso estimators, the properties of the proposed shrinkage averaging estimators resulting from these strategies are explored by means of Monte-Carlo studies and are compared to traditional approaches where the tuning parameter is simply selected via cross validation criteria. The results show that the proposed SAE methodology can improve an estimators’ overall performance and reveal and incorporate tuning parameter uncertainty. As an illustration, selected methods are applied to some recent data from a study on leadership behavior in life science companies.     

A comparison of some robust,adaptive, and partially adaptive estimators of regression models

Numerous estimation techniques for regression models have been proposed. These procedures differ in how sample information is used in the estimation procedure. The efficiency of least squares (OLS) estimators implicity assumes normally distributed residuals and is very sensitive to departures from normality, particularly to "outliers" and thick-tailed distributions. Lead absolute deviation (LAD) estimators are less sensitive to outliers and are optimal for laplace random disturbances, but not for normal errors. This paper reports monte carlo comparisons of OLS,LAD, two robust estimators discussed by huber, three partially adaptiveestimators, newey's generalized method of moments estimator, and an adaptive maximum likelihood estimator based on a normal kernal studied by manski. This paper is the first to compare the relative performance of some adaptive robust estimators (partially adaptive and adaptive procedures) with some common nonadaptive robust estimators. The partially adaptive estimators are based on three flxible parametric distributions for the errors. These include the power exponential (Box-Tiao) and generalized t distributions, as well as a distribution for the errors, which is not necessarily symmetric. The adaptive procedures are "fully iterative" rather than one step estimators. The adaptive estimators have desirable large sample properties, but these properties do not necessarily carry over to the small sample case.

The monte carlo comparisons of the alternative estimators are based on four different specifications for the error distribution: a normal, a mixture of normals (or variance-contaminated normal), a bimodal mixture of normals, and a lognormal. Five hundred samples of 50 are used. The adaptive and partially adaptive estimators perform very well relative to the other estimation procedures considered, and preliminary results suggest that in some important cases they can perform much better than OLS with 50 to 80% reductions in standard errors.

     

A comparison of some robust, adaptive, and partially adaptive estimators of regression models

Numerous estimation techniques for regression models have been proposed. These procedures differ in how sample information is used in the estimation procedure. The efficiency of least squares (OLS) estimators implicity assumes normally distributed residuals and is very sensitive to departures from normality, particularly to "outliers" and thick-tailed distributions. Lead absolute deviation (LAD) estimators are less sensitive to outliers and are optimal for laplace random disturbances, but not for normal errors. This paper reports monte carlo comparisons of OLS,LAD, two robust estimators discussed by huber, three partially adaptiveestimators, newey's generalized method of moments estimator, and an adaptive maximum likelihood estimator based on a normal kernal studied by manski. This paper is the first to compare the relative performance of some adaptive robust estimators (partially adaptive and adaptive procedures) with some common nonadaptive robust estimators. The partially adaptive estimators are based on three flxible parametric distributions for the errors. These include the power exponential (Box-Tiao) and generalized t distributions, as well as a distribution for the errors, which is not necessarily symmetric. The adaptive procedures are "fully iterative" rather than one step estimators. The adaptive estimators have desirable large sample properties, but these properties do not necessarily carry over to the small sample case.

The monte carlo comparisons of the alternative estimators are based on four different specifications for the error distribution: a normal, a mixture of normals (or variance-contaminated normal), a bimodal mixture of normals, and a lognormal. Five hundred samples of 50 are used. The adaptive and partially adaptive estimators perform very well relative to the other estimation procedures considered, and preliminary results suggest that in some important cases they can perform much better than OLS with 50 to 80% reductions in standard errors.     

Logistic回归的双层变量选择研究

变量选择是统计建模的重要环节,选择合适的变量可以建立结构简单、预测精准的稳健模型。本文在logistic回归下提出了新的双层变量选择惩罚方法——adaptive Sparse Group Lasso(adSGL),其独特之处在于基于变量的分组结构作筛选,实现了组内和组间双层选择。该方法的优点是对各单个系数和组系数采取不同程度的惩罚,避免了过度惩罚大系数,从而提高了模型的估计和预测精度。求解的难点是惩罚似然函数不是严格凸的,因此本文基于组坐标下降法求解模型,并建立了调整参数的选取准则。模拟分析表明,对比现有代表性方法Sparse Group Lasso、Group Lasso及Lasso,adSGL法不仅提高了双层选择精度,而且降低了模型误差。最后本文将adSGL法应用到信用卡信用评分研究,对比logistic回归,它具有更高的分类精度和稳健性。     

Variable Selection via Regression Trees in the Presence of Irrelevant Variables

Many tree algorithms have been developed for regression problems. Although they are regarded as good algorithms, most of them suffer from loss of prediction accuracy when there are many irrelevant variables and the number of predictors exceeds the number of observations. We propose the multistep regression tree with adaptive variable selection to handle this problem. The variable selection step and the fitting step comprise the multistep method.

The multistep generalized unbiased interaction detection and estimation (GUIDE) with adaptive forward selection (fg) algorithm, as a variable selection tool, performs better than some of the well-known variable selection algorithms such as efficacy adaptive regression tube hunting (EARTH), FSR (false selection rate), LSCV (least squares cross-validation), and LASSO (least absolute shrinkage and selection operator) for the regression problem. The results based on simulation study show that fg outperforms other algorithms in terms of selection result and computation time. It generally selects the important variables correctly with relatively few irrelevant variables, which gives good prediction accuracy with less computation time.     

Bi-level variable selection via adaptive sparse group Lasso

Penalization has been extensively adopted for variable selection in regression. In some applications, covariates have natural grouping structures, where those in the same group have correlated measurements or related functions. Under such settings, variable selection should be conducted at both the group-level and within-group-level, that is, a bi-level selection. In this study, we propose the adaptive sparse group Lasso (adSGL) method, which combines the adaptive Lasso and adaptive group Lasso (GL) to achieve bi-level selection. It can be viewed as an improved version of sparse group Lasso (SGL) and uses data-dependent weights to improve selection performance. For computation, a block coordinate descent algorithm is adopted. Simulation shows that adSGL has satisfactory performance in identifying both individual variables and groups and lower false discovery rate and mean square error than SGL and GL. We apply the proposed method to the analysis of a household healthcare expenditure data set.