Berry-Esseen Bounds for Projection Parameters and Partial Correlations with Increasing Dimension

The linear regression model can be used even when the true regression function is not linear. The resulting estimated linear function is the best linear approximation to the regression function and the vector $\beta$ of the coefficients of this linear approximation are the projection parameter. We provide finite sample bounds on the Normal approximation to the law of the least squares estimator of the projection parameters normalized by the sandwich-based standard error. Our results hold in the increasing dimension setting and under minimal assumptions on the distribution of the response variable. Furthermore, we construct confidence sets for $\beta$ in the form of hyper-rectangles and establish rates on their coverage accuracy. We provide analogous results for partial correlations among the entries of sub-Gaussian vectors.