Thin Sets Are Not Equally Thin: Minimax Learning of Submanifold Integrals
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915838326472704 |
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| author | Chen, Xiaohong Gao, Wayne Yuan |
| author_facet | Chen, Xiaohong Gao, Wayne Yuan |
| contents | Many economic parameters are identified by ``thin sets'' (submanifolds with Lebesgue measure zero) and hence difficult to recover from data in an ambient space. This paper provides a unified theory for estimation and inference of such ``thin-set'' identified functionals. We show that thin sets are \emph{not} equally thin: their intrinsic dimensionality $m$ matters in a precise manner. For a nonparametric regression $h_0$ with Hölder smoothness $s$ and $d$-dimensional covariates in the ambient space, we show that $n^{-\frac{s}{2s+d-m}}$ is the minimax optimal rate of estimating linear and nonlinear (e.g., quadratic, upper contour) integrals of $h_0$ on an $m$-dimensional submanifold ($0\leq m < d$), which is the fastest possible attainable rate among all estimators. The minimax lower bound rate result is generalized to estimating submanifold integrals when $h_0$ is a nonparametric density and a nonparametric instrumental variable function. The asymptotic normality of t statistics is established via sieve Riesz representation, and the corresponding inference is computed using Sobol points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_12673 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Thin Sets Are Not Equally Thin: Minimax Learning of Submanifold Integrals Chen, Xiaohong Gao, Wayne Yuan Econometrics Many economic parameters are identified by ``thin sets'' (submanifolds with Lebesgue measure zero) and hence difficult to recover from data in an ambient space. This paper provides a unified theory for estimation and inference of such ``thin-set'' identified functionals. We show that thin sets are \emph{not} equally thin: their intrinsic dimensionality $m$ matters in a precise manner. For a nonparametric regression $h_0$ with Hölder smoothness $s$ and $d$-dimensional covariates in the ambient space, we show that $n^{-\frac{s}{2s+d-m}}$ is the minimax optimal rate of estimating linear and nonlinear (e.g., quadratic, upper contour) integrals of $h_0$ on an $m$-dimensional submanifold ($0\leq m < d$), which is the fastest possible attainable rate among all estimators. The minimax lower bound rate result is generalized to estimating submanifold integrals when $h_0$ is a nonparametric density and a nonparametric instrumental variable function. The asymptotic normality of t statistics is established via sieve Riesz representation, and the corresponding inference is computed using Sobol points. |
| title | Thin Sets Are Not Equally Thin: Minimax Learning of Submanifold Integrals |
| topic | Econometrics |
| url | https://arxiv.org/abs/2507.12673 |