Christoph Breunig, Xiaohong Chen
arXiv 28 Jan 2021 · Mathematics — Statistics Theory · publishedSpringer proceedings in mathematics & statistics (2023) · 2 citations (OpenAlex)
arXiv:2101.12282 · PDF · DOI · OpenAlex · Extracted main text
This paper considers adaptive, minimax estimation of a quadratic functional in a nonparametric instrumental variables (NPIV) model, which is an important problem in optimal estimation of a nonlinear functional of an ill-posed inverse regression with an unknown operator. We first show that a leave-one-out, sieve NPIV estimator of the quadratic functional can attain a convergence rate that coincides with the lower bound previously derived in Chen and Christensen [2018]. The minimax rate is achieved by the optimal choice of the sieve dimension (a key tuning parameter) that depends on the smoothness of the NPIV function and the degree of ill-posedness, both are unknown in practice. We next propose a Lepski-type data-driven choice of the key sieve dimension adaptive to the unknown NPIV model features. The adaptive estimator of the quadratic functional is shown to attain the minimax optimal rate in the severely ill-posed case and in the regular mildly ill-posed case, but up to a multiplicative $\sqrt{\log n}$ factor in the irregular mildly ill-posed case.
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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 | X. Chen and T. M. Christensen (2018) Optimal sup-norm rates and uniform inference on nonlinear functionals of nonparametric iv regression | 1.000 | 10 | 3 | 100% |
| 2 | R. Blundell, X. Chen, and D. Kristensen (2007) Semi-nonparametric iv estimation of shape-invariant engel curves | 0.874 | 5 | 2 | 100% |
| 3 | X. Chen, T. Christensen, and S. Kankanala (2021) Adaptive estimation and uniform confidence bands for nonparametric iv | 0.851 | 13 | 6 | 62% |
| 4 | C. Breunig and J. Johannes (2016) Adaptive estimation of functionals in nonparametric instrumental regression | 0.737 | 3 | 2 | 100% |
| 5 | S. Efromovich and M. Low (1996) On optimal adaptive estimation of a quadratic functional | 0.737 | 3 | 2 | 100% |
| 6 | C. Breunig and X. Chen (2021) Adaptive, rate-optimal hypothesis testing in nonparametric iv models | 0.659 | 7 | 3 | 29% |
| 7 | O. V. Lepski (1990) On a problem of adaptive estimation in gaussian white noise | 0.644 | 2 | 2 | 100% |
| 8 | O. Collier, L. Comminges, and A. B. Tsybakov (2017) Minimax estimation of linear and quadratic functionals on sparsity classes | 0.644 | 2 | 2 | 100% |
| 9 | O. V. Lepski and V. G. Spokoiny (1997) Optimal pointwise adaptive methods in nonparametric estimation | 0.644 | 2 | 2 | 100% |
| 10 | O. V. Lepski, E. Mammen, and V. G. Spokoiny (1997) Optimal spatial adaptation to inhomogeneous smoothness: an approach based on kernel estimates with variable bandwidth selectors | 0.644 | 2 | 2 | 100% |
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