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      A Residual Bootstrap for High-Dimensional Regression with Near Low-Rank Designs


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          We study the residual bootstrap (RB) method in the context of high-dimensional linear regression. Specifically, we analyze the distributional approximation of linear contrasts \(c^{\top} (\hat{\beta}_{\rho}-\beta)\), where \(\hat{\beta}_{\rho}\) is a ridge-regression estimator. When regression coefficients are estimated via least squares, classical results show that RB consistently approximates the laws of contrasts, provided that \(p\ll n\), where the design matrix is of size \(n\times p\). Up to now, relatively little work has considered how additional structure in the linear model may extend the validity of RB to the setting where \(p/n\asymp 1\). In this setting, we propose a version of RB that resamples residuals obtained from ridge regression. Our main structural assumption on the design matrix is that it is nearly low rank --- in the sense that its singular values decay according to a power-law profile. Under a few extra technical assumptions, we derive a simple criterion for ensuring that RB consistently approximates the law of a given contrast. We then specialize this result to study confidence intervals for mean response values \(X_i^{\top} \beta\), where \(X_i^{\top}\) is the \(i\)th row of the design. More precisely, we show that conditionally on a Gaussian design with near low-rank structure, RB simultaneously approximates all of the laws \(X_i^{\top}(\hat{\beta}_{\rho}-\beta)\), \(i=1,\dots,n\). This result is also notable as it imposes no sparsity assumptions on \(\beta\). Furthermore, since our consistency results are formulated in terms of the Mallows (Kantorovich) metric, the existence of a limiting distribution is not required.

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