Sparse group lasso for multiclass functional logistic regression models

Sparsity-inducing penalties are useful tools for variable selection and are also effective for regression problems where the data are functions. We consider the problem of selecting not only variables but also decision boundaries in multiclass logistic regression models for functional data, using sparse regularization. The parameters of the functional logistic regression model are estimated in the framework of the penalized likelihood method with the sparse group lasso-type penalty, and then tuning parameters for the model are selected using the model selection criterion. The effectiveness of the proposed method is investigated through simulation studies and the analysis of a gene expression data set.     

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.     

Model Selection and Shrinkage: An Overview

This special issue is concerned with model selection and shrinkage estimators. This Introduction gives an overview of the papers published in this special issue.     

Oracle Inequalities for Convex Loss Functions with Nonlinear Targets

This article considers penalized empirical loss minimization of convex loss functions with unknown target functions. Using the elastic net penalty, of which the Least Absolute Shrinkage and Selection Operator (Lasso) is a special case, we establish a finite sample oracle inequality which bounds the loss of our estimator from above with high probability. If the unknown target is linear, this inequality also provides an upper bound of the estimation error of the estimated parameter vector. Next, we use the non-asymptotic results to show that the excess loss of our estimator is asymptotically of the same order as that of the oracle. If the target is linear, we give sufficient conditions for consistency of the estimated parameter vector. We briefly discuss how a thresholded version of our estimator can be used to perform consistent variable selection. We give two examples of loss functions covered by our framework.     

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.     

Hard thresholding regression

In this paper, we propose the hard thresholding regression (HTR) for estimating high‐dimensional sparse linear regression models. HTR uses a two‐stage convex algorithm to approximate the ?₀‐penalized regression: The first stage calculates a coarse initial estimator, and the second stage identifies the oracle estimator by borrowing information from the first one. Theoretically, the HTR estimator achieves the strong oracle property over a wide range of regularization parameters. Numerical examples and a real data example lend further support to our proposed methodology.     

The Frisch–Waugh–Lovell theorem for the lasso and the ridge regression

The Frisch–Waugh–Lovell (FWL) (partitioned regression) theorem is essential in regression analysis. This is partly because it is quite useful to derive theoretical results. The lasso regression and the ridge regression, both of which are penalized least-squares regressions, have become popular statistical techniques. This article describes that the FWL theorem remains valid for these penalized least-squares regressions. More precisely, we demonstrate that the covariates corresponding to unpenalized regression parameters in these penalized least-squares regression can be projected out. Some other results related to the FWL theorem in such penalized least-squares regressions are also presented.     

Semi-parametric segmentation of multiple series using a DP-Lasso strategy

We consider a semi-parametric approach to perform the joint segmentation of multiple series sharing a common functional part. We propose an iterative procedure based on Dynamic Programming for the segmentation part and Lasso estimators for the functional part. Our Lasso procedure, based on the dictionary approach, allows us to both estimate smooth functions and functions with local irregularity, which permits more flexibility than previous proposed methods. This yields to a better estimation of the functional part and improvements in the segmentation. The performance of our method is assessed using simulated data and real data from agriculture and geodetic studies. Our estimation procedure results to be a reliable tool to detect changes and to obtain an interpretable estimation of the functional part of the model in terms of known functions.     

Shrinkage estimation of proportion via logit penalty

By releasing the unbiasedness condition, we often obtain more accurate estimators due to the bias–variance trade-off. In this paper, we propose a class of shrinkage proportion estimators which show improved performance over the sample proportion. We provide the “optimal” amount of shrinkage. The advantage of the proposed estimators is given theoretically as well as explored empirically by simulation studies and real data analyses.     

Regularized proportional odds models

The proportional odds model (POM) is commonly used in regression analysis to predict the outcome for an ordinal response variable. The maximum likelihood estimation (MLE) approach is typically used to obtain the parameter estimates. The likelihood estimates do not exist when the number of parameters, p, is greater than the number of observations n. The MLE also does not exist if there are no overlapping observations in the data. In a situation where the number of parameters is less than the sample size but p is approaching to n, the likelihood estimates may not exist, and if they exist they may have quite large standard errors. An estimation method is proposed to address the last two issues, i.e. complete separation and the case when p approaches n, but not the case when p>n. The proposed method does not use any penalty term but uses pseudo-observations to regularize the observed responses by downgrading their effect so that they become close to the underlying probabilities. The estimates can be computed easily with all commonly used statistical packages supporting the fitting of POMs with weights. Estimates are compared with MLE in a simulation study and an application to the real data.