Maximum Likelihood for Variance Estimation in High-Dimensional Linear Models

We study maximum likelihood estimators (MLEs) for the residual variance, the signal-to-noise ratio, and other variance parameters in high-dimensional linear models. These parameters are essential in many statistical applications involving regression diagnostics, inference, tuning parameter selection for high-dimensional regression, and other applications, including genetics. The estimators that we study are not new, and have been widely used for variance component estimation in linear random-effects models. However, our analysis is new and it implies that the MLEs, which were devised for random-effects models, may also perform very well in high-dimensional linear models with fixed-effects, which are more commonly studied in some areas of high-dimensional statistics. The MLEs are shown to be consistent and asymptotically normal in fixed-effects models with random design, in asymptotic settings where the number of predictors ($p$) is proportional to the number of observations ($n$). Moreover, the estimators’ asymptotic variance can be given explicitly in terms moments of the Marcenko-Pastur distribution. A variety of analytical and empirical results show that the MLEs outperform other, previously proposed estimators for variance parameters in high-dimensional linear models with fixed-effects. More broadly, the results in this paper illustrate a strategy for drawing connections between fixed- and random-effects models in high dimensions, which may be useful in other applications.