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Stein's method for the matrix normal distribution

Robert E. Gaunt, Frédéric Ouimet, Donald Richards

Abstract

This work presents the first systematic development of Stein's method for matrix distributions. We establish the basic essential ingredients of Stein's method for matrix normal approximation: we derive a generator-based Stein identity from a matrix Ornstein--Uhlenbeck diffusion with two-sided scales, provide an explicit semigroup representation for the solution of the Stein equation, and obtain regularity estimates for the solution. The new methodology is illustrated with three statistical applications, these being smooth Wasserstein distance bounds to quantify the matrix central limit theorem, a Wasserstein distance bound for the matrix normal approximation of the centered matrix $T$ distribution, and the derivation of Stein's method-of-moments estimators for scale parameters of the matrix normal distribution.

Stein's method for the matrix normal distribution

Abstract

This work presents the first systematic development of Stein's method for matrix distributions. We establish the basic essential ingredients of Stein's method for matrix normal approximation: we derive a generator-based Stein identity from a matrix Ornstein--Uhlenbeck diffusion with two-sided scales, provide an explicit semigroup representation for the solution of the Stein equation, and obtain regularity estimates for the solution. The new methodology is illustrated with three statistical applications, these being smooth Wasserstein distance bounds to quantify the matrix central limit theorem, a Wasserstein distance bound for the matrix normal approximation of the centered matrix distribution, and the derivation of Stein's method-of-moments estimators for scale parameters of the matrix normal distribution.
Paper Structure (16 sections, 11 theorems, 185 equations)

This paper contains 16 sections, 11 theorems, 185 equations.

Table of Contents

  1. Introduction
  2. Definitions and notation
  3. Main results
  4. Applications
  5. Comparison between the standardized sample mean of iid matrix-valued observations and the matrix normal distribution with the same covariance
  6. Comparison between the centered matrix T and normal distributions
  7. Estimation of the parameters of a centered matrix normal distribution
  8. Proofs
  9. Proof of Proposition \ref{['prop:OU.process.generator']}
  10. Proof of Proposition \ref{['prop:OU.process.distribution']}
  11. Proof of Theorem \ref{['thm:Stein.solutions']}
  12. Proof of Theorem \ref{['thm:smoothness.estimates']}
  13. Proof of Proposition \ref{['prop:generalization.Gaunt.2013.Example.2.6']}
  14. Proof of Proposition \ref{['prop:T.vs.normal']}
  15. Proof of Proposition \ref{['prop:projected.Stein']}
  16. ...and 1 more sections

Key Result

Proposition 3.1

For any $f\in C^2(\mathbb{R}^{\nu\times d})$, we have where $\nabla = (\partial / \partial X_{ij})_{1\leq i \leq \nu, 1 \leq j \leq d}$ denotes the $\nu\times d$ matrix of first-order partial derivatives.

Theorems & Definitions (18)

  • Proposition 3.1: Infinitesimal generator
  • Proposition 3.2
  • Corollary 3.3: Stein characterization
  • Theorem 3.4: Stein solutions for $\alpha$-Hölder test functions
  • Theorem 3.5: Regularity of the solution of the Stein equation
  • Remark 3.6
  • Proposition 4.1
  • Proposition 4.2: Wasserstein distance bound between the matrix $T$ and matrix normal
  • Remark 4.3
  • Remark 4.4
  • ...and 8 more