Abstract
The paper considers a linear regression model in high-dimension for which the predictive variables can change the influence on the response variable at unknown times (called changepoints). Moreover, the particular case of the heavy-tailed errors is considered. In this case, least square method with LASSO or adaptive LASSO penalty can not be used since the theoretical assumptions do not occur or the estimators are not robust. Then, the quantile model with SCAD penalty or median regression with LASSO-type penalty allows, in the same time, to estimate the parameters on every segment and eliminate the irrelevant variables. We show that, for the two penalized estimation methods, the oracle properties is not affected by the change-point estimation. Convergence rates of the estimators for the change-points and for the regression parameters, by the two methods are found. Monte-Carlo simulations illustrate the performance of the methods.
Highlights
A model which changes at some observations is called a change-point model
Recall for a median regression in high dimension the paper of Wang,[4] where a L1 penalized least absolute deviation method is considered, when the overall variable number is larger than the observation number
In order to study the properties of the penalized estimators in a model with breaking, we need corresponding results obtained without change-points when τ = 1/2: by (Ref. 9), (Ref. 3) and for a some τ ∈ (0, 1) by (Ref. 2)
Summary
A model which changes at some observations is called a change-point model. The location of these changes (called change-points, breaks, changes) may be known or unknown. Recall for a median regression in high dimension the paper of Wang,[4] where a L1 penalized least absolute deviation method is considered, when the overall variable number is larger than the observation number. In a multiple change-point model, the break estimation could affects the estimator properties This is the main interest of this paper. A change-point linear model in high-dimension was considered by Ciuperca but under stronger assumptions that the errors have mean zero and bounded variance[5]. For both methods, the oracle properties and convergence rate of the estimators are obtained.
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