# Shrinking characteristics of precision matrix estimators

## TL;DR

This paper introduces a flexible framework for shrinking specific characteristics of precision matrix estimators, improving predictive performance even if the estimators are less accurate for the true matrix.

## Contribution

It develops a new class of precision matrix estimators with a penalty on chosen characteristics, providing convergence guarantees and an efficient algorithm for computation.

## Key findings

- Estimators outperform competitors in predictive tasks.
- Better prediction performance despite worse population matrix estimation.
- Convergence rate bounds established for the proposed estimators.

## Abstract

We propose a framework to shrink a user-specified characteristic of a precision matrix estimator that is needed to fit a predictive model. Estimators in our framework minimize the Gaussian negative loglikelihood plus an $L_1$ penalty on a linear or affine function evaluated at the optimization variable corresponding to the precision matrix. We establish convergence rate bounds for these estimators and propose an alternating direction method of multipliers algorithm for their computation. Our simulation studies show that our estimators can perform better than competitors when they are used to fit predictive models. In particular, we illustrate cases where our precision matrix estimators perform worse at estimating the population precision matrix but better at prediction.