A nonparametric regression alternative to empirical Bayes approaches to simultaneous estimation

TL;DR

This paper introduces a purely frequentist nonparametric regression method for simultaneous estimation, addressing issues of Bayesian reliance and unreliable density estimates in empirical Bayes approaches.

Contribution

It proposes a novel frequentist alternative linking simultaneous estimation to penalized nonparametric regression with shape constraints, avoiding Bayesian assumptions.

Findings

Achieves asymptotically optimal regret.

Performs competitively or better than empirical Bayes methods in simulations.

Effective in analyzing spatial gene expression data.

Abstract

The simultaneous estimation of multiple unknown parameters lies at heart of a broad class of important problems across science and technology. Currently, the state-of-the-art performance in the such problems is achieved by nonparametric empirical Bayes methods. However, these approaches still suffer from two major issues. First, they solve a frequentist problem but do so by following Bayesian reasoning, posing a philosophical dilemma that has contributed to somewhat uneasy attitudes toward empirical Bayes methodology. Second, their computation relies on certain density estimates that become extremely unreliable in some complex simultaneous estimation problems. In this paper, we study these issues in the context of the canonical Gaussian sequence problem. We propose an entirely frequentist alternative to nonparametric empirical Bayes methods by establishing a connection between simultaneous estimation and penalized nonparametric regression. We use flexible regularization strategies, such as shape constraints, to derive accurate estimators without appealing to Bayesian arguments. We prove that our estimators achieve asymptotically optimal regret and show that they are competitive with or can outperform nonparametric empirical Bayes methods in simulations and an analysis of spatially resolved gene expression data.