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Gaussian quasi-likelihood analysis for non-Gaussian linear mixed-effects model with system noise

Takumi Imamura, Hiroki Masuda

TL;DR

This work develops a Gaussian quasi-likelihood framework for non-Gaussian linear mixed-effects models with integrated OU system noise under unbalanced longitudinal designs. It establishes uniform convergence of a quasi-KL divergence, derives quasi-scores and quasi-observed information, and proves root-$N$ consistency and asymptotic normality for the joint GQMLE, including a tail-probability bound. A three-stage stepwise GQMLE is proposed, shown to be first-order equivalent to the joint estimator with explicit second-order differences, offering substantial computational savings. Numerical experiments confirm the theoretical results and demonstrate favorable performance of the stepwise method in large samples and under non-Gaussian random effects, with practical guidance for inference and model selection.

Abstract

We consider statistical inference for a class of mixed-effects models with system noise described by a non-Gaussian integrated Ornstein-Uhlenbeck process. Under the asymptotics where the number of individuals goes to infinity with possibly unbalanced sampling frequency across individuals, we prove some theoretical properties of the Gaussian quasi-likelihood function, followed by the asymptotic normality and the tail-probability estimate of the associated estimator. In addition to the joint inference, we propose and investigate the three-stage inference strategy, revealing that they are first-order equivalent while quantitatively different in the second-order terms. Numerical experiments are given to illustrate the theoretical results.

Gaussian quasi-likelihood analysis for non-Gaussian linear mixed-effects model with system noise

TL;DR

This work develops a Gaussian quasi-likelihood framework for non-Gaussian linear mixed-effects models with integrated OU system noise under unbalanced longitudinal designs. It establishes uniform convergence of a quasi-KL divergence, derives quasi-scores and quasi-observed information, and proves root- consistency and asymptotic normality for the joint GQMLE, including a tail-probability bound. A three-stage stepwise GQMLE is proposed, shown to be first-order equivalent to the joint estimator with explicit second-order differences, offering substantial computational savings. Numerical experiments confirm the theoretical results and demonstrate favorable performance of the stepwise method in large samples and under non-Gaussian random effects, with practical guidance for inference and model selection.

Abstract

We consider statistical inference for a class of mixed-effects models with system noise described by a non-Gaussian integrated Ornstein-Uhlenbeck process. Under the asymptotics where the number of individuals goes to infinity with possibly unbalanced sampling frequency across individuals, we prove some theoretical properties of the Gaussian quasi-likelihood function, followed by the asymptotic normality and the tail-probability estimate of the associated estimator. In addition to the joint inference, we propose and investigate the three-stage inference strategy, revealing that they are first-order equivalent while quantitatively different in the second-order terms. Numerical experiments are given to illustrate the theoretical results.

Paper Structure

This paper contains 19 sections, 6 theorems, 97 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Setup and objective
  4. Outline
  5. Comments on model selection
  6. Basic notation
  7. Joint Gaussian quasi-likelihood analysis
  8. Uniform convergence of quasi-Kullback-Leibler divergence
  9. Quasi-score function
  10. Quasi-observed information
  11. Asymptotic normality and tail-probability estimate
  12. Stepwise Gaussian quasi-likelihood analysis
  13. Construction and asymptotics
  14. Proof of Theorem \ref{['ti:QLA.thm2']}
  15. Tail-probability estimate
  16. ...and 4 more sections

Key Result

Lemma 2.6

Let $\Theta\subset\mathbb{R}^p$ be a bounded convex domain, $q>p\vee 2$, and let $\chi_{Ni}(\theta): \Theta\to\mathbb{R}$, $i\le N$, $N\ge 1$, be random functions. Then, we have If in particular $(\partial_\theta^k \chi_{Ni}(\theta))_{i=1}^N$ for $k\in\{0,1\}$ and $\theta\in\Theta$ forms a martingale difference array with respect to some filtration $(\mathcal{F}_{Ni})_{i\le N}$, then

Figures (3)

  • Figure 1: The box plot of the computation loads for calculating the joint GQMLE and the stepwise GQMLE in $N = 1000$ for $1000$ iterations.
  • Figure 2: Histograms of the studentized joint and stepwise GQMLEs and probability density function of standard Gaussian distribution (red curve)
  • Figure 3: Normal Q-Q plots of the studentized joint and stepwise GQMLEs

Theorems & Definitions (14)

  • Remark 2.4
  • Lemma 2.6
  • proof
  • Theorem 2.9
  • Lemma 2.10
  • proof
  • proof : Proof of Theorem \ref{['ti:QLA.thm1']}
  • Remark 2.11
  • Corollary 2.12
  • Remark 2.13: Gaussian case
  • ...and 4 more