Sparse linear regression methods for high-dimensional data often assume that residuals have constant variance. When this assumption is violated, it can lead to bias in estimated coefficients, prediction intervals with improper length, and increased type I errors. This paper proposes a heteroscedastic (H) high-dimensional linear regression model through a partitioned empirical Bayes Expectation Conditional Maximization (H-PROBE) algorithm. H-PROBE is a computationally efficient maximum a posteriori (MAP) estimation approach based on a Parameter-Expanded Expectation-Conditional-Maximization (PX-ECM) algorithm. It requires minimal prior assumptions on the regression parameters through plug-in empirical Bayes estimates of hyperparameters. The variance model uses recent advances in multivariate log-Gamma distribution theory and can include covariates hypothesized to impact heterogeneity. The motivation of our approach is a study relating Aphasia Quotient (AQ) to high-resolution T2 neuroimages of brain damage in stroke patients. AQ is a vital measure of language impairment and informs treatment decisions, but it is challenging to measure and subject to heteroscedastic errors. As a result, it is of clinical importance -- and the goal of this paper -- to use high-dimensional neuroimages to predict and provide prediction intervals for AQ that accurately reflect the heterogeneity in the residual variance. Our analysis demonstrates that H-PROBE can use markers of heterogeneity to provide prediction interval widths that are narrower than standard methods without sacrificing coverage. Further, through extensive simulation studies, we exhibit that the proposed approach results in superior prediction, variable selection, and predictive inference than competing methods.

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