Jelena Bradic, Victor Chernozhukov, Whitney K. Newey, Yinchu Zhu
arXiv 27 Dec 2019 · Mathematics — Statistics Theory · 4 citations (OpenAlex)
arXiv:1912.12213 · PDF · DOI · OpenAlex · Extracted main text
Estimating linear, mean-square continuous functionals is a pivotal challenge in statistics. In high-dimensional contexts, this estimation is often performed under the assumption of exact model sparsity, meaning that only a small number of parameters are precisely non-zero. This excludes models where linear formulations only approximate the underlying data distribution, such as nonparametric regression methods that use basis expansion such as splines, kernel methods or polynomial regressions. Many recent methods for root-$n$ estimation have been proposed, but the implications of exact model sparsity remain largely unexplored. In particular, minimax optimality for models that are not exactly sparse has not yet been developed. This paper formalizes the concept of approximate sparsity through classical semi-parametric theory. We derive minimax rates under this formulation for a regression slope and an average derivative, finding these bounds to be substantially larger than those in low-dimensional, semi-parametric settings. We identify several new phenomena. We discover new regimes where rate double robustness does not hold, yet root-$n$ estimation is still possible. In these settings, we propose an estimator that achieves minimax optimal rates. Our findings further reveal distinct optimality boundaries for ordered versus unordered nonparametric regression estimation.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Javanmard, A. and Montanari, A (2018) Debiasing the lasso: Optimal sample size for gaussian designs | 1.000 | 13 | 7 | 100% |
| 2 | Cai, T. T. and Guo, Z (2017) Confidence intervals for high-dimensional linear regression: Minimax rates and adaptivity | 1.000 | 11 | 8 | 100% |
| 3 | Chernozhukov, V., Chetverikov, D., Demirer, M., Duflo, E., Hansen, C… (2018) Double/debiased machine learning for treatment and structural parameters self | 1.000 | 9 | 4 | 100% |
| 4 | Belloni, A., Chernozhukov, V., and Hansen, C (2014) Inference on treatment effects after selection among high-dimensional controls self | 1.000 | 8 | 3 | 100% |
| 5 | Javanmard, A. and Montanari, A (2014) Confidence intervals and hypothesis testing for high-dimensional regression | 1.000 | 7 | 5 | 100% |
| 6 | Van de Geer, S., Bühlmann, P., Ritov, Y., and Dezeure, R (2014) On asymptotically optimal confidence regions and tests for high-dimensional models | 1.000 | 7 | 5 | 100% |
| 7 | Hirshberg, D. A. and Wager, S (2021) Augmented minimax linear estimation | 1.000 | 7 | 4 | 100% |
| 8 | Tsybakov, A. B (2009) Introduction to Nonparametric Estimation | 1.000 | 7 | 3 | 100% |
| 9 | Chernozhukov, V., Newey, W. K., and Singh, R (2018) Learning l2 continuous regression functionals via regularized riesz representers self | 1.000 | 5 | 5 | 100% |
| 10 | Athey, S., Imbens, G. W., and Wager, S (2018) Approximate residual balancing: De-biased inference of average treatment effects in high dimensions | 1.000 | 5 | 4 | 100% |
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