A pivotal transform for the high-dimensional location-scale model

TL;DR

This paper introduces a generalized transformation of the log-likelihood for high-dimensional linear models with known noise distribution up to scale, enabling scale-independent tuning and improved estimation.

Contribution

It proposes a new transformation extending the square root Lasso, allowing tuning parameter selection without knowledge of the scale parameter in high-dimensional models.

Findings

The transformation generalizes the square root Lasso for quadratic loss.

The method achieves oracle inequalities and variable selection consistency.

It provides asymptotic efficiency for scale and intercept estimators.

Abstract

We study the high-dimensional linear model with noise distribution known up to a scale parameter. With an $\ell_1$-penalty on the regression coefficients, we show that a transformation of the log-likelihood allows for a choice of the tuning parameter not depending on the scale parameter. This transformation is a generalization of the square root Lasso for quadratic loss. The tuning parameter can asymptotically be taken at the detection edge. We establish an oracle inequality, variable selection and asymptotic efficiency of the estimator of the scale parameter and the intercept. The examples include Subbotin distributions and the Gumbel distribution.