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

Xiaohong Chen, Wayne Yuan Gao

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 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.

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

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 matters in a precise manner. For a nonparametric regression with Hölder smoothness and -dimensional covariates in the ambient space, we show that is the minimax optimal rate of estimating linear and nonlinear (e.g., quadratic, upper contour) integrals of on an -dimensional submanifold (), which is the fastest possible attainable rate among all estimators. The minimax lower bound rate result is generalized to estimating submanifold integrals when 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.

Paper Structure

This paper contains 40 sections, 22 theorems, 386 equations, 8 tables.

Table of Contents

  1. Introduction
  2. Organization of the Paper
  3. Setup and Examples
  4. Setup
  5. Motivating Examples
  6. Category 1:
  7. Category 2:
  8. Minimax Rate of Estimation: Lower Bound
  9. Case 1: $h_0$ is a Regression or a Density
  10. For Linear Functional $\theta_0=L(h_0)$
  11. For Nonlinear Functional $\theta_0=\Gamma(h_0)$
  12. Case 2: When $h_0$ is an NPIV Structural Function
  13. For Linear Functional $\theta_0=L(h_0)$
  14. For Nonlinear Functional $\theta_0=\Gamma(h_0)$
  15. Rate-Optimal Estimation
  16. ...and 25 more sections

Key Result

Theorem 1

Under Assumptions assu:RegLevelSet, assu:Jacob, assu:density_x, assu:w and assu:e2_below, the rate $r^*_{n}=n^{-\frac{s}{2s+d-m}}$ is the minimax rate lower bound for the estimation of $\theta_{0}\left(P,w\right):=\int_{{\cal M}}h_{0}\left(x\right)w\left(x\right)d{\cal H}^{m}\left(x\right)$, i.e. where $P$ is any joint probability distribution of $\left(X_{i},Y_{i}\right)$ that satisfies $h_{0}\l

Theorems & Definitions (66)

  • Example 1: Maximum Score Estimation of Binary Choice Models
  • Example 2: Optimal Linear Treatment Assignment
  • Example 3: Aggregate Parameter over Estimated Subpopulation
  • Example 4: Average Treatment Effects under Propensity Score or Density Trimming
  • Example 5: ReLU-Based Generalized Maximum Score Estimator under Multi-Index Single-Crossing Conditions
  • Example 6: Generalized Partial Means and Nonparametric Regression with Generated Covariates
  • Example 7: Marginal Treatment Effects and Policy Relevant Treatment Effects
  • Example 8: Weighted Average Derivatives
  • Example 9: Structural Functions in NPIV Regression
  • Example 10: Nonparametric Quantile
  • ...and 56 more