A note on the approximate admissibility of regularized estimators in the Gaussian sequence model

TL;DR

This paper extends the concept of approximate admissibility from constrained least squares estimators to all convex penalized estimators in the Gaussian sequence model, providing simplified proofs and explicit bounds.

Contribution

It generalizes previous results by proving convex penalized estimators are approximately admissible and offers clearer proofs with explicit bounds.

Findings

Convex penalized estimators are approximately admissible in the Gaussian sequence model.

Simplified proof techniques for approximate admissibility.

Explicit bounds for the universal constant in the admissibility concept.

Abstract

We study the problem of estimating an unknown vector $\theta$ from an observation $X$ drawn according to the normal distribution with mean $\theta$ and identity covariance matrix under the knowledge that $\theta$ belongs to a known closed convex set $\Theta$. In this general setting, Chatterjee (2014) proved that the natural constrained least squares estimator is "approximately admissible" for every $\Theta$. We extend this result by proving that the same property holds for all convex penalized estimators as well. Moreover, we simplify and shorten the original proof considerably. We also provide explicit upper and lower bounds for the universal constant underlying the notion of approximate admissibility.