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A sliced Wasserstein and diffusion approach to random coefficient models

Keunwoo Lim, Ting Ye, Fang Han

TL;DR

This work develops a scalable minimum-distance estimator for the distribution of random coefficients in a linear random coefficient model by integrating the sliced Wasserstein distance with a k-NN weighting scheme. It introduces a Monte Carlo approximation and two practical optimization algorithms, proving estimation consistency and deriving convergence rates under both bounded and unbounded covariate regimes; it also connects the framework to diffusion-process based methods via a regularized functional and to causal inference through a regularized working model. The diffusion-based approach yields a continuous-time evolution for the coefficient distribution with a particle-system algorithm that achieves polynomial-time complexity under reasonable scaling. Empirical results on simulations demonstrate computational efficiency and accuracy, while an application to causal inference with ACTG 175 data provides distributional treatment-effect insights that support established clinical conclusions. Overall, the paper offers a novel, scalable, and theoretically grounded framework for estimating and utilizing the distribution of random coefficients in high-dimensional settings, with clear links to diffusion modeling and causal analysis.

Abstract

We propose a new minimum-distance estimator for linear random coefficient models. This estimator integrates the recently advanced sliced Wasserstein distance with the nearest neighbor methods, both of which enhance computational efficiency. We demonstrate that the proposed method is consistent in approximating the true distribution. Moreover, our formulation naturally leads to a diffusion process-based algorithm and is closely connected to treatment effect distribution estimation -- both of which are of independent interest and hold promise for broader applications.

A sliced Wasserstein and diffusion approach to random coefficient models

TL;DR

This work develops a scalable minimum-distance estimator for the distribution of random coefficients in a linear random coefficient model by integrating the sliced Wasserstein distance with a k-NN weighting scheme. It introduces a Monte Carlo approximation and two practical optimization algorithms, proving estimation consistency and deriving convergence rates under both bounded and unbounded covariate regimes; it also connects the framework to diffusion-process based methods via a regularized functional and to causal inference through a regularized working model. The diffusion-based approach yields a continuous-time evolution for the coefficient distribution with a particle-system algorithm that achieves polynomial-time complexity under reasonable scaling. Empirical results on simulations demonstrate computational efficiency and accuracy, while an application to causal inference with ACTG 175 data provides distributional treatment-effect insights that support established clinical conclusions. Overall, the paper offers a novel, scalable, and theoretically grounded framework for estimating and utilizing the distribution of random coefficients in high-dimensional settings, with clear links to diffusion modeling and causal analysis.

Abstract

We propose a new minimum-distance estimator for linear random coefficient models. This estimator integrates the recently advanced sliced Wasserstein distance with the nearest neighbor methods, both of which enhance computational efficiency. We demonstrate that the proposed method is consistent in approximating the true distribution. Moreover, our formulation naturally leads to a diffusion process-based algorithm and is closely connected to treatment effect distribution estimation -- both of which are of independent interest and hold promise for broader applications.

Paper Structure

This paper contains 19 sections, 14 theorems, 154 equations, 3 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Method
  3. Algorithm
  4. Theory
  5. Assumptions
  6. Estimation
  7. Computation
  8. A diffusion process approach
  9. Application to causal inference
  10. Numerical experiments
  11. Simulation studies
  12. Causal RCM
  13. Proofs
  14. Proofs of Assumptions
  15. Proofs of results concerning estimation
  16. ...and 4 more sections

Key Result

Proposition 4.1

Suppose there exist constants $C_{f} > 0$ and $\kappa > 1$ such that $X = (1, X_{2:d})$ and the density $f_{X_{2:d}}$ of $X_{2:d}$ satisfies: for all $T \in \mathbb{R}^{d-1}$. Assumption assumption: unbounded covariates then holds with $\alpha = 1/(\kappa + 1)$, $\tau_{0} = \mathcal{S}(\mathbb{S}^{d-1})C_{f}/2^{3\kappa + 3}$, and $\rho_{0} = 1/4$, where $\mathcal{S}(\mathbb{S}^{d-1})$ is a surfac

Figures (3)

  • Figure 1: Coefficient samples and outputs of Algorithms \ref{['algorithm: bcd']}-\ref{['algorithm: flow']}.
  • Figure 2: Outputs of Algorithm \ref{['algorithm: flow']} with varying numbers of particles and regularization parameter.
  • Figure 3: Treatment coefficient samples for Therapies 2–4.

Theorems & Definitions (29)

  • Remark 2.1
  • Proposition 4.1
  • Remark 4.1
  • Theorem 4.1
  • Theorem 4.2
  • Proposition 4.2
  • Proposition 4.3
  • Theorem 4.3
  • Proposition 5.1: Theorem 2, liutkus2019sliced
  • Remark 5.1
  • ...and 19 more