论文标题
两级混合模型的数值集成的渐近学
Asymptotics of numerical integration for two-level mixed models
论文作者
论文摘要
我们研究具有单个分组因子的混合模型，其中关于未知参数的推断需要优化由棘手的积分定义的边际似然。低维数值整合技术通常用于近似这些积分，并根据所得的近似边缘可能性推断参数。对于满足显式规律条件的一般混合模型，我们得出当使用自适应数值整合以近似边际可能性时，可能会导致可能性和最大似然估计量产生的随机相对错误率。然后，我们将分析专门针对具有指数式家庭响应和多元高斯随机效应的良好指定的广义线性混合模型，从而验证了规律性条件是否存在，因此收敛速率适用。我们还证明，对于满足非常弱浓度条件的可能性的模型，从边缘可能性的非自适应数值整合近似值中的最大似然估计器是不一致的，进一步激励了自适应数值整合，作为混合模型中推断的首选工具。在本文中复制模拟的代码在https://github.com/awstringer1/aq-weory-paper-code上提供。
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.
