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.

翻译：我们研究具有单一分组因子的混合模型，其中对未知参数的推断需要优化一个由难以处理的积分定义的边际似然。低维数值积分技术通常被用来近似这些积分，而关于参数的推断则基于由此得到的近似边际似然。对于满足明确正则性条件的一类通用混合模型，我们推导了当使用自适应数值积分来近似边际似然时，似然函数和最大似然估计量所承受的随机相对误差率。随后，我们将分析专门应用于具有指数族响应和多变量高斯随机效应的、设定正确的广义线性混合模型，验证了正则性条件成立，从而证明了收敛率适用。我们还证明了，对于似然函数满足极弱集中条件的模型，基于非自适应数值积分近似的边际似然所得的最大似然估计量是不一致的，这进一步促使自适应数值积分成为混合模型推断的首选工具。重现本文中模拟的代码可在 https://github.com/awstringer1/aq-theory-paper-code 获取。