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Global-local shrinkage priors for modeling random effects in multivariate spatial small area estimation

Shushi Nishina, Takahiro Onizuka, Shintaro Hashimoto

Abstract

Small area estimation (SAE) plays a central role in survey statistics and epidemiology, providing reliable estimates for domains with limited sample sizes. The multivariate Fay-Herriot model has been extensively used for this purpose, because it enhances estimation accuracy by borrowing strength across multiple correlated variables. In this paper, we develop a Bayesian extension of the multivariate Fay-Herriot model that enables flexible, component-specific shrinkage of the random effects. The proposed approach employs global-local priors formulated through a sandwich mixture representation, allowing adaptive regularization of each element of the random-effect vectors. This construction yields greater robustness and prevents excessive shrinkage in areas exhibiting strong underlying signals. In addition, we incorporate spatial dependence into the model to account for geographical correlation across small areas. The resulting spatial multivariate framework simultaneously exploits cross-variable relationships and spatial structure, yielding improved estimation efficiency. The utility of the proposed method is demonstrated through simulation studies and an empirical application to real survey data.

Global-local shrinkage priors for modeling random effects in multivariate spatial small area estimation

Abstract

Small area estimation (SAE) plays a central role in survey statistics and epidemiology, providing reliable estimates for domains with limited sample sizes. The multivariate Fay-Herriot model has been extensively used for this purpose, because it enhances estimation accuracy by borrowing strength across multiple correlated variables. In this paper, we develop a Bayesian extension of the multivariate Fay-Herriot model that enables flexible, component-specific shrinkage of the random effects. The proposed approach employs global-local priors formulated through a sandwich mixture representation, allowing adaptive regularization of each element of the random-effect vectors. This construction yields greater robustness and prevents excessive shrinkage in areas exhibiting strong underlying signals. In addition, we incorporate spatial dependence into the model to account for geographical correlation across small areas. The resulting spatial multivariate framework simultaneously exploits cross-variable relationships and spatial structure, yielding improved estimation efficiency. The utility of the proposed method is demonstrated through simulation studies and an empirical application to real survey data.
Paper Structure (19 sections, 1 theorem, 22 equations, 10 figures, 1 table)

This paper contains 19 sections, 1 theorem, 22 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Multivariate FH model and sparsity for random effects
  3. Global-local priors based on sandwich-mixture covariance structure for multivariate spatial FH models
  4. The proposed methods
  5. Prior specification
  6. Posterior computation
  7. Non-Gaussian response
  8. Simulation
  9. Setting
  10. Simulation results
  11. Real data analysis
  12. Dataset and setting
  13. Results
  14. Discussion
  15. Proof
  16. ...and 4 more sections

Key Result

Proposition 3.1

Under the prior distribution proposal, if the priors of $\mathbf{\Sigma}$, $\rho \in (0,1)$, $\tau$ and $\lambda_{ij}$ are proper, then the posterior density is proper.

Figures (10)

  • Figure 1: Results from a single simulation run for true random effects (left) and posterior mean estimates of the local parameters under HS (center) and SpaHS (right) for Scenario 5 with $m=500$ and observation variance case (a). For HS, each area has a single local parameter, and thus the $x$-axis represents the area index. In contrast, SpaHS assigns two local parameters to each area, which are plotted in two dimensions.
  • Figure 2: The average absolute deviation (AAD), average squared deviation (ASD), coverage probability of the 95% credible interval and average length of the 95% credible interval under the variance scenario (a). The figure displays the median computed over 50 replications and the horizontal axis is the number of small areas.
  • Figure 3: The average absolute deviation (AAD), average squared deviation (ASD), coverage probability of the 95% credible interval and average length of the 95% credible interval under the variance scenario (b). The figure displays the median computed over 50 replications and the horizontal axis is the number of small areas.
  • Figure 4: Variance on the log-scale of median household income and poverty rate of direct estimates for 413 counties in West Census Region, based on 5-year period estimates from the 2022 American Community Survey.
  • Figure 5: Posterior means of small area mean $\bm{\theta}_i$ for the SpaHS, SpaGa and SpaFH methods.
  • ...and 5 more figures

Theorems & Definitions (2)

  • Proposition 3.1: Posterior propriety
  • proof : Proof of Proposition \ref{['propriety']}