Thin Sets Are Not Equally Thin: Minimax Learning of Submanifold Integrals

Xiaohong Chen and Wayne Yuan Gao
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Xiaohong Chen: Yale University
Wayne Yuan Gao: University of Pennsylvania

No 2450R1, Cowles Foundation Discussion Papers from Cowles Foundation for Research in Economics, Yale University

Abstract: 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 not equally thin: their intrinsic dimensionality m matters in a precise manner. For a nonparametric regression h0 with Hšlder smoothness s and d-dimensional covariates in the ambient space, we show that n^-s/(2s+d-m) is the minimax optimal rate of estimating linear and nonlinear (e.g., quadratic, upper contour) integrals of h0 on an m-dimensional submanifold (0

Pages: 106 pages
Date: 2026-03-05
New Economics Papers: this item is included in nep-ecm
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