Structural Change in Sparsity

In the high-dimensional sparse modeling literature, it has been crucially assumed that the sparsity structure of the model is homogeneous over the entire population. That is, the identities of important regressors are invariant across the population and across the individuals in the collected sample. In practice, however, the sparsity structure may not always be invariant in the population, due to heterogeneity across different sub-populations. We consider a general, possibly non-smooth M-estimation framework, allowing a possible structural change regarding the identities of important regressors in the population. Our penalized M-estimator not only selects covariates but also discriminates between a model with homogeneous sparsity and a model with a structural change in sparsity. As a result, it is not necessary to know or pretest whether the structural change is present, or where it occurs. We derive asymptotic bounds on the estimation loss of the penalized M-estimators, and achieve the oracle properties. We also show that when there is a structural change, the estimator of the threshold parameter is super-consistent. If the signal is relatively strong, the rates of convergence can be further improved and asymptotic distributional properties of the estimators including the threshold estimator can be established using an adaptive penalization. The proposed methods are then applied to quantile regression and logistic regression models and are illustrated via Monte Carlo experiments.

The Lasso for High-Dimensional Regression with a Possible Change-Point

We consider a high-dimensional regression model with a possible change-point due to a covariate threshold and develop the Lasso estimator of regression coefficients as well as the threshold parameter. Our Lasso estimator not only selects covariates but also selects a model between linear and threshold regression models. Under a sparsity assumption, we derive non-asymptotic oracle inequalities for both the prediction risk and the $ell_1$ estimation loss for regression coefficients. Since the Lasso estimator selects variables simultaneously, we show that oracle inequalities can be established without pretesting the existence of the threshold effect. Furthermore, we establish conditions under which the estimation error of the unknown threshold parameter can be bounded by a nearly $n^{-1}$ factor even when the number of regressors can be much larger than the sample size ($n$). We illustrate the usefulness of our proposed estimation method via Monte Carlo simulations and an application to real data.