Inference of random effects for linear mixed-effects models with a fixed number of clusters

Selection of linear mixed‐effects models for clustered data

“…Note that conditions (A.I)–(A.IV) are also imposed by Chang et al (2022) to establish the asymptotic behaviors of the ML estimators for models , which lay an important foundation of the theoretical development for selecting mixed‐effects models in this article. The techniques developed in Chang et al (2022) focus on decomposing the log‐likelihood functions of candidate models for parameter estimation.…”

“…Note that conditions (A.I)–(A.IV) are also imposed by Chang et al (2022) to establish the asymptotic behaviors of the ML estimators for models , which lay an important foundation of the theoretical development for selecting mixed‐effects models in this article. The techniques developed in Chang et al (2022) focus on decomposing the log‐likelihood functions of candidate models for parameter estimation. The decomposition is mainly constituted by many quadratic forms of $$$ {\boldsymbol{H}}^{-1}\left(\gamma, \boldsymbol{\theta} \right) $$$ given in Lemmas 1–3 in Data S1.…”

“…More precisely, the CKL loss/risk and the conditional log‐likelihood functions are constituted by many quadratic forms of $$$ {\boldsymbol{H}}^{-1}\left(\gamma, \boldsymbol{\theta} \right) $$$ and $$$ {\boldsymbol{H}}^{-2}\left(\gamma, \boldsymbol{\theta} \right) $$$. Besides the asymptotic properties of the aforementioned quadratic forms developed in Chang et al (2022), the asymptotic properties of other quadratic forms are established in Lemmas 4–6 in Data S1.…”

“…We further decompose $$$ \mathrm{E}{\left\Vert \hat{\boldsymbol{S}}\left(\alpha, \gamma; \boldsymbol{\theta} \right)-\mathrm{E}\Big(\boldsymbol{S}|\boldsymbol{y}\Big)\right\Vert}^2 $$$ into three parts, which are useful in establishing asymptotic properties for model selection. The following proposition, together with theorems 1 and 3–5 of Chang et al (2022), further provides an innovative way to circumvent the difficulty in analyzing the key components of the likelihood function (for details, see Propositions 2 and 3.…”

“…Another main reason is that the model parameters are not consistently estimable when $$$ m $$$ is fixed, which further hinders us from inferring linear mixed‐effects models through EBLUPs based on a finite number of clusters. However, Chang et al (2022) develop the asymptotic theory of maximum likelihood (ML) estimators of variance parameters for linear mixed‐effects models when models are correctly or incorrectly specified and $$$ m $$$ is allowed to be fixed, where a novel decomposition of the inverse covariance matrix is introduced, which also lays the foundation for the theoretical development of selecting linear mixed‐effects models. In addition, as shown in an example in Chang et al (2022), the EBLUPs of random effects continue to outperform the least‐squares predictors (obtained by treating the random effects as fixed effects) even when $$$ m $$$ does not go to infinity with the sample size.…”