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

Estimation of signal-to-noise ratios and residual variances in high-dimensional linear models has various important applications, including heritability estimation in bioinformatics. One widely used estimator is the Gaussian random-effects maximum likelihood estimator (MLE), based on the likelihood of the homogeneous Gaussian random-effects model in which both the regression coefficients and the noise variables are assumed to be i.i.d. Gaussian. This paper studies the behavior of this likelihood estimator under model misspecification. For isotropic random designs with independent, symmetric, sub-Gaussian entries, we establish consistency and asymptotic normality of the SNR MLE for fixed dense coefficient vectors and independent, centered, heteroscedastic finite-moment noise, allowing moderately heavy-tailed errors. We also give parallel consistency and central limit results for correlated Gaussian noise as a benchmark. The asymptotic variance depends on the limiting aspect ratio, the true SNR, and a scalar noise-square fluctuation parameter. This explicit form yields feasible plug-in confidence intervals under independent noise in two cases where the fluctuation parameter can be estimated from response fourth moments: heterogeneous Gaussian noise and homogeneous non-Gaussian noise. Numerical simulations compare likelihood-based and method-of-moments confidence intervals under heterogeneous and non-Gaussian noise, and a real-data illustration demonstrates the resulting calibrations on high-dimensional text features.