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Local Prediction-Powered Inference

Yanwu Gu, Dong Xia

Abstract

To infer a function value on a specific point $x$, it is essential to assign higher weights to the points closer to $x$, which is called local polynomial / multivariable regression. In many practical cases, a limited sample size may ruin this method, but such conditions can be improved by the Prediction-Powered Inference (PPI) technique. This paper introduced a specific algorithm for local multivariable regression using PPI, which can significantly reduce the variance of estimations without enlarge the error. The confidence intervals, bias correction, and coverage probabilities are analyzed and proved the correctness and superiority of our algorithm. Numerical simulation and real-data experiments are applied and show these conclusions. Another contribution compared to PPI is the theoretical computation efficiency and explainability by taking into account the dependency of the dependent variable.

Local Prediction-Powered Inference

Abstract

To infer a function value on a specific point , it is essential to assign higher weights to the points closer to , which is called local polynomial / multivariable regression. In many practical cases, a limited sample size may ruin this method, but such conditions can be improved by the Prediction-Powered Inference (PPI) technique. This paper introduced a specific algorithm for local multivariable regression using PPI, which can significantly reduce the variance of estimations without enlarge the error. The confidence intervals, bias correction, and coverage probabilities are analyzed and proved the correctness and superiority of our algorithm. Numerical simulation and real-data experiments are applied and show these conclusions. Another contribution compared to PPI is the theoretical computation efficiency and explainability by taking into account the dependency of the dependent variable.
Paper Structure (23 sections, 6 theorems, 85 equations, 7 figures, 1 table)

This paper contains 23 sections, 6 theorems, 85 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Works
  3. Multivariable Local Linear Regression
  4. Prediction-Powered Inference
  5. Theory
  6. Preliminaries
  7. Conventional Estimation
  8. Local Prediction-Powered Inference Estimator
  9. Confidence Interval and Hypothesis Test
  10. Coverage Probability and Bias Correction
  11. High Dimensional Condition and Limited-sample Condition
  12. Experiments
  13. Numerical Simulation
  14. House Price Inference
  15. Air Quality Inference
  16. ...and 8 more sections

Key Result

Theorem 1

Under Assumption asump:multi, for $h=n^{-\beta}, 0<\beta<p^{-1}$ as $n\to\infty$, the conditional bias of local linear regression and derivative given by the solution of Equation equ:abexp have the asymptotic expansions as where The covariance of estimation can be described as where $J_i=\int u_1^iK^2(u)du$.

Figures (7)

  • Figure 1: PPI Improvement Outline
  • Figure 2: Error Scatter Plot
  • Figure 3: Error of Gradient Estimation
  • Figure 4: Fitted Normal Distribution Comparison
  • Figure 5: Deduction of Mean Absolute Error
  • ...and 2 more figures

Theorems & Definitions (12)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • proof
  • proof
  • proof
  • proof
  • ...and 2 more