Asymptotics of numerical integration for two-level mixed models
File(s)
Accepted version
Author(s)
Bilodeau, Blair
Stringer, Alex
Tang, Yanbo
Type
Journal Article
Abstract
We study mixed models with a single grouping factor where inference about unknown parameters requires optimizing a marginal likelihood defined by an intractable integral. Low-dimensional numerical integration techniques are regularly used to approximate these integrals with inferences about parameters based on the resulting approximate marginal likelihood. For a generic class of mixed models that satisfy explicit regularity conditions we derive the stochastic relative error rate incurred for both the likelihood and maximum likelihood estimator when adaptive numerical integration is used to approximate the marginal likelihood. We then specialize the analysis to well-specified generalized linear mixed models having exponential family response and multivariate Gaussian random effects, verifying that the regularity conditions hold and hence that the convergence rates apply. We also prove that for models with likelihoods satisfying very weak concentration conditions that the maximum likelihood estimators from non-adaptive numerical integration approximations of the marginal likelihood are not consistent, further motivating adaptive numerical integration as the preferred tool for inference in mixed models. Code to reproduce the simulations in this paper is provided at https://github.com/awstringer1/aq-theory-paper-code.
Date Issued
2026-11-01
Date Acceptance
2025-04-11
Citation
Bernoulli: a journal of mathematical statistics and probability, 2026, 32 (4), pp.2594-2618
ISSN
1350-7265
Publisher
Bernoulli Society for Mathematical Statistics and Probability
Start Page
2594
End Page
2618
Journal / Book Title
Bernoulli: a journal of mathematical statistics and probability
Volume
32
Issue
4
Copyright Statement
Copyright © 2026 ISI/BS.
Publication Status
Published
Date Publish Online
2026-07-30