The Linear Lasso: a location model resolution

TL;DR

This paper introduces a location model approach to the Lasso problem, simplifying variable selection and inference by focusing on the response variable and enabling efficient, one-dimensional analysis.

Contribution

It develops a novel location model methodology for Lasso that simplifies inference and variable removal, providing a new perspective on variable selection.

Findings

Inference is primarily one-dimensional.

Focus is on the response variable, not the least squares estimate.

Efficient computation and scalar marginal models are achievable.

Abstract

We use location model methodology to guide the least squares analysis of the Lasso problem of variable selection and inference. The nuisance parameter is taken to be an indicator for the selection of explanatory variables and the interest parameter is the response variable itself. Recent theory eliminates the nuisance parameter by marginalization on the data space and then uses the resulting distribution for inference concerning the interest parameter. We develop this approach and find: that primary inference is essentially one-dimensional rather than $n$-dimensional; that inference focuses on the response variable itself rather than the least squares estimate (as variables are removed); that first order probabilities are available; that computation is relatively easy; that a scalar marginal model is available; and that ineffective variables can be removed by distributional tilt or shift.