Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Hierarchical Bayes justification of Empirical Bayes Confidence Intervals

Aditi Sen, Masayo Y. Hirose, Partha Lahiri

TL;DR

This work addresses interval estimation for random effects in a two-level Fay–Herriot-type hierarchy by linking EB confidence intervals with HB posteriors through a carefully constructed, area-specific matching prior on the hyperparameter $A$. The authors derive an explicit prior $oldsymbol{ extpi}(A)$ that ensures posterior coverage $P^{oldsymbol{ extpi}}( heta_i q I_i | y)=1- ext{α} + o_p(m^{-1})$, while also proving posterior propriety under mild conditions. Through simulations and a real-data baseball example, they show that the EB intervals $I_i^{ ext{N}}$ and $I_i^{ ext{YL}}$ achieve second-order accuracy in both EB and posterior senses, with comparable lengths and robust performance across settings. Overall, the paper provides a principled reconciliation of EB and HB approaches for small-area interval estimation and highlights the value of area-specific, data-driven priors for improving interval performance in hierarchical models.

Abstract

Multi-level normal hierarchical models, also interpreted as mixed effects models, play an important role in developing statistical theory in multi-parameter estimation for a wide range of applications. In this article, we propose a novel reconciliation framework of the empirical Bayes (EB) and hierarchical Bayes approaches for interval estimation of random effects under a two-level normal model. Our framework shows that a second-order efficient empirical Bayes confidence interval, with EB coverage error of order $O(m^{-3/2})$, $m$ being the number of areas in the area-level model, can also be viewed as a credible interval whose posterior coverage is close to the nominal level, provided a carefully chosen prior - referred to as a 'matching prior' - is placed on the hyperparameters. While existing literature has examined matching priors that reconcile frequentist and Bayesian inference in various settings, this paper is the first to study matching priors with the goal of interval estimation of random effects in a two-level model. We obtain an area-dependent matching prior on the variance component that achieves a proper posterior under mild regularity conditions. The theoretical results in the paper are corroborated through a Monte Carlo simulation study and a real data analysis.

On the Hierarchical Bayes justification of Empirical Bayes Confidence Intervals

TL;DR

This work addresses interval estimation for random effects in a two-level Fay–Herriot-type hierarchy by linking EB confidence intervals with HB posteriors through a carefully constructed, area-specific matching prior on the hyperparameter . The authors derive an explicit prior that ensures posterior coverage , while also proving posterior propriety under mild conditions. Through simulations and a real-data baseball example, they show that the EB intervals and achieve second-order accuracy in both EB and posterior senses, with comparable lengths and robust performance across settings. Overall, the paper provides a principled reconciliation of EB and HB approaches for small-area interval estimation and highlights the value of area-specific, data-driven priors for improving interval performance in hierarchical models.

Abstract

Multi-level normal hierarchical models, also interpreted as mixed effects models, play an important role in developing statistical theory in multi-parameter estimation for a wide range of applications. In this article, we propose a novel reconciliation framework of the empirical Bayes (EB) and hierarchical Bayes approaches for interval estimation of random effects under a two-level normal model. Our framework shows that a second-order efficient empirical Bayes confidence interval, with EB coverage error of order , being the number of areas in the area-level model, can also be viewed as a credible interval whose posterior coverage is close to the nominal level, provided a carefully chosen prior - referred to as a 'matching prior' - is placed on the hyperparameters. While existing literature has examined matching priors that reconcile frequentist and Bayesian inference in various settings, this paper is the first to study matching priors with the goal of interval estimation of random effects in a two-level model. We obtain an area-dependent matching prior on the variance component that achieves a proper posterior under mild regularity conditions. The theoretical results in the paper are corroborated through a Monte Carlo simulation study and a real data analysis.

Paper Structure

This paper contains 30 sections, 15 theorems, 119 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Existing literature
  3. Interval estimation of small area means
  4. Matching priors in SAE
  5. Our contribution
  6. Notations and regularity conditions
  7. Asymptotic expansion of posterior coverage
  8. Desired prior and its propriety
  9. Matching prior
  10. Propriety of matching prior
  11. Special cases
  12. Two-level model without covariates
  13. Balanced case of the two-level model
  14. Simulation Study
  15. Simulation setting
  16. ...and 15 more sections

Key Result

Theorem 1

Suppose the regularity conditions R$1$-R$5$ hold. Then the posterior coverage of $I_{i}^\text{N}$ under M$2$ is given by, where $c_i$ is a function of $(y, z,D_i)$ given by

Figures (3)

  • Figure 1: Plot of true batting averages ($\theta_i$) and its three EBCIs: $I_i^{\text{YL}}$ (M3), $I_i^\text{N}$ (M4) and $I_i^{\text{YL}}$ (M4), for $m=18$ players.
  • Figure 2: Plot of PC and Monte Carlo error for three EBCIs: $I_i^{\text{YL}}$ (M3), $I_i^\text{N}$ (M4) and $I_i^{\text{YL}}$ (M4), for $m=18$ players.
  • Figure 3: Plot of estimates of $A$ for the three EBCIs : $\tilde{A}_{i;\text{sp}}$ for $I_i^\text{YL}$ (M3), $\tilde{A}_{i}$ for $I_i^\text{N}$ (M4) and $\hat{A}_{i}$ for $I_i^\text{YL}$ (M4).

Theorems & Definitions (32)

  • Theorem 1
  • Remark 4.1
  • Remark 4.2
  • Proposition 1
  • Remark 4.3
  • Remark 4.4
  • Theorem 2
  • Remark 5.1
  • Theorem 3
  • Proposition 2
  • ...and 22 more