# Maximum Likelihood for Variance Estimation in High-Dimensional Linear Models

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Lee H. Dicker,
Murat A. Erdogdu
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Proceedings of the 19th International Conference on Artificial Intelligence and Statistics, PMLR 51:159-167, 2016.

#### Abstract

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.

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