Post-Regularization Confidence Bands for High Dimensional Nonparametric Models with Local Sparsity
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We propose a novel high dimensional nonparametric model named ATLAS which naturally generlizes the sparse additive model. Given a covariate of interest X_j, the ATLAS model assumes the mean function can be locally approximated by a sparse additive function whose sparsity pattern may vary from the global perspective. We propose to infer the marginal influence function f_j^*(z) = E[f(X_1,..., X_d) | X_j = z] using a new kernel-sieve approach that combines the local kernel regression with the B-spline basis approximation. We prove the rate of convergence for estimating f_j^* under the supremum norm. We also propose two types of confidence bands for f_j^* and illustrate their statistical-comptuational tradeoffs. Thorough numerical results on both synthetic data and real-world genomic data are provided to demonstrate the efficacy of the theory.
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