Document Type

Discussion Paper

Publication Date

3-2026

CFDP Number

2450R1

CFDP Pages

104

Journal of Economic Literature (JEL) Code(s)

N/A

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 Hölder 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 ≤ 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 h0 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.

Download

Included in

Economics Commons

Share

COinS