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Data-Driven Uniform Inference for General Continuous Treatment Models via Minimum-Variance Weighting

Chunrong Ai, Wei Huang, Zheng Zhang

TL;DR

This work develops a nonparametric framework for estimating a general dose-response function (GDRF) under continuous treatment with unconfoundedness, using minimum-variance covariate-balancing weights and a weighted local linear regression. It provides a fully data-driven inference pipeline, including closed-form minimum-variance weights via sieving, a weighted bootstrap for uniform confidence bands, and two tuning strategies (undersmoothing and Lepski) to ensure valid band coverage. The authors establish Uniform Bahadur representations and prove the validity of the uniform confidence bands, supported by extensive simulations and an empirical study. The approach yields robust, model-misspecification-resistant inference for a wide class of dose-response objects, such as the average dose-response function (ADRF), quantile dose-response function (QDRF), and other expectiles.

Abstract

Ai et al. (2021) studied the estimation of a general dose-response function (GDRF) of a continuous treatment that includes the average dose-response function, the quantile dose-response function, and other expectiles of the dose-response distribution. They specified the GDRF as a parametric function of the treatment status only and proposed a weighted regression with the weighting function estimated using the maximum entropy approach. This paper specifies the GDRF as a nonparametric function of the treatment status, proposes a weighted local linear regression for estimating GDRF, and develops a bootstrap procedure for constructing the uniform confidence bands. We propose stable weights with minimum sample variance while eliminating the sample association between the treatment and the confounding variables. The proposed weights admit a closed-form expression, allowing them to be computed efficiently in the bootstrap sampling. Under certain conditions, we derive the uniform Bahadur representation for the proposed estimator of GDRF and establish the validity of the corresponding uniform confidence bands. A fully data-driven approach to choosing the undersmooth tuning parameters and a data-driven bias-control confidence band are included. A simulation study and an application demonstrate the usefulness of the proposed approach.

Data-Driven Uniform Inference for General Continuous Treatment Models via Minimum-Variance Weighting

TL;DR

This work develops a nonparametric framework for estimating a general dose-response function (GDRF) under continuous treatment with unconfoundedness, using minimum-variance covariate-balancing weights and a weighted local linear regression. It provides a fully data-driven inference pipeline, including closed-form minimum-variance weights via sieving, a weighted bootstrap for uniform confidence bands, and two tuning strategies (undersmoothing and Lepski) to ensure valid band coverage. The authors establish Uniform Bahadur representations and prove the validity of the uniform confidence bands, supported by extensive simulations and an empirical study. The approach yields robust, model-misspecification-resistant inference for a wide class of dose-response objects, such as the average dose-response function (ADRF), quantile dose-response function (QDRF), and other expectiles.

Abstract

Ai et al. (2021) studied the estimation of a general dose-response function (GDRF) of a continuous treatment that includes the average dose-response function, the quantile dose-response function, and other expectiles of the dose-response distribution. They specified the GDRF as a parametric function of the treatment status only and proposed a weighted regression with the weighting function estimated using the maximum entropy approach. This paper specifies the GDRF as a nonparametric function of the treatment status, proposes a weighted local linear regression for estimating GDRF, and develops a bootstrap procedure for constructing the uniform confidence bands. We propose stable weights with minimum sample variance while eliminating the sample association between the treatment and the confounding variables. The proposed weights admit a closed-form expression, allowing them to be computed efficiently in the bootstrap sampling. Under certain conditions, we derive the uniform Bahadur representation for the proposed estimator of GDRF and establish the validity of the corresponding uniform confidence bands. A fully data-driven approach to choosing the undersmooth tuning parameters and a data-driven bias-control confidence band are included. A simulation study and an application demonstrate the usefulness of the proposed approach.
Paper Structure (19 sections, 4 theorems, 78 equations, 2 figures, 6 tables)

This paper contains 19 sections, 4 theorems, 78 equations, 2 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Basic Framework
  3. Minimum-Variance Weights and Uniform Inference Algorithm
  4. Theoretical Properties
  5. Uniform Bahadur Representations
  6. Validity of Uniform Confidence Bands
  7. Practical Details
  8. Basis functions
  9. Data-driven tuning parameters and confidence bands
  10. Pilot parameters
  11. Method 1 (Undersmoothing): Data-driven undersmooth bandwidth
  12. Method 2 (Lepski): Data-driven confidence bands
  13. Validity of the Lepski method
  14. Numerical Studies
  15. Simulation Study
  16. ...and 4 more sections

Key Result

Theorem 1

Under Assumptions as:TYindep-ass:h, we obtain the linear (Bahadur) representations for $\widehat{g}_h(t)$ and $\widehat{g'}_h(t)$ uniformly over $t$: and For $\widehat{g}^{(b)}$ and $\widehat{g'}^{(b)}$, for $b=1,\ldots,B$, we have: and Moreover, we show that and for $b=1,\ldots,B$,

Figures (2)

  • Figure 1: Plots of the estimators (dotted) of $g$ or $g^{'}$, with the 90% uniform confidence band (dash-dotted) for $g$ and for testing $H_0:g^{'}=0$ (dashed), from samples with $N=1200$. The true $g^{'}$ curves are depicted by solid lines (Row 2).
  • Figure 2: Plots of the estimated QDRF with 95% uniform confidence band ($q = 0.5$ and 0.75 from bottom to top) using Methods 1 (top left) and 2 (bottom left), the estimated first derivative of the QDRF with 95% confidence band (2nd and 3rd column) using Methods 1 (top) and 2 (bottom), and the plots of $d_{L_\infty}(\widehat{g}_{h_j},\widehat{g}_{h_{j-1}})$ (top right) and $d_{L_\infty}(\widehat{\partial_t g}_{h_j},\widehat{\partial_t g}_{h_{j-1}})$ (bottom right) for Method 1, where $g$ is the median dose-response function.

Theorems & Definitions (9)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 1
  • Remark 5
  • Corollary 1
  • Theorem 2
  • Theorem 3