Misspecification Analysis of High-Dimensional Random Effects Models for Estimation of Signal-to-Noise Ratios

TL;DR

This paper investigates the consistency and distribution of REML estimators for signal-to-noise ratios in high-dimensional linear models, accounting for heteroscedastic and correlated noise, with theoretical and numerical validation.

Contribution

It establishes the asymptotic properties of REML estimators under more realistic noise conditions in high-dimensional settings, relaxing some previous assumptions.

Findings

REML estimator is consistent under heteroscedastic, correlated noise.

Asymptotic distribution of REML is derived for fixed coefficients.

Numerical simulations support theoretical results and suggest assumptions may be relaxed.

Abstract

Estimation of signal-to-noise ratios and residual variances in high-dimensional linear models has various important applications including, e.g. heritability estimation in bioinformatics. One commonly used estimator, usually referred to as REML, is based on the likelihood of the random effects model, in which both the regression coefficients and the noise variables are respectively assumed to be i.i.d Gaussian random variables. In this paper, we aim to establish the consistency and asymptotic distribution of the REML estimator for the SNR, when the actual coefficient vector is fixed, and the actual noise is heteroscedastic and correlated, at the cost of assuming the entries of the design matrix are independent and skew-free. The asymptotic variance can be also consistently estimated when the noise is heteroscedastic but uncorrelated. Extensive numerical simulations illustrate our theoretical findings and also suggest some assumptions imposed in our theoretical results are likely relaxable.