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

TL;DR

This paper develops a unified minimax theory for estimating functionals on thin sets, such as submanifolds, revealing how their intrinsic dimension affects estimation rates in nonparametric models.

Contribution

It introduces the first minimax optimal rates for estimating integrals over submanifolds, accounting for their intrinsic dimension, and extends the theory to density and instrumental variable functions.

Findings

Minimax rate for submanifold integral estimation depends on intrinsic dimension m.

Establishes asymptotic normality of t-statistics using sieve Riesz representation.

Provides inference methods using Sobol points for thin-set functionals.

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 \emph{not} equally thin: their intrinsic dimensionality $m$ matters in a precise manner. For a nonparametric regression $h_0$ with H\"{o}lder 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.