Variance function estimation in high-dimensions
This addresses a gap in high-dimensional regression for applications like biostatistics and finance where non-constant error variances are common but often ignored.
The paper tackles the problem of estimating both mean and variance functions in high-dimensional heteroscedastic regression, where error variances are non-constant, and shows that their proposed HIPPO estimator achieves the oracle property with convergence rates matching those of a known true model.
We consider the high-dimensional heteroscedastic regression model, where the mean and the log variance are modeled as a linear combination of input variables. Existing literature on high-dimensional linear regres- sion models has largely ignored non-constant error variances, even though they commonly occur in a variety of applications ranging from biostatis- tics to finance. In this paper we study a class of non-convex penalized pseudolikelihood estimators for both the mean and variance parameters. We show that the Heteroscedastic Iterative Penalized Pseudolikelihood Optimizer (HIPPO) achieves the oracle property, that is, we prove that the rates of convergence are the same as if the true model was known. We demonstrate numerical properties of the procedure on a simulation study and real world data.