Mixed domain asymptotics for a stochastic process model with time trend and measurement error

Abstract

We consider a stochastic process model with time trend and measurement error. We establish consistency and derive the limiting distributions of the maximum likelihood (ML) estimators of the covariance function parameters under a general asymptotic framework, including both the fixed domain and the increasing domain frameworks, even when the time trend model is misspecified or its complexity increases with the sample size. In particular, the convergence rates of the ML estimators are thoroughly characterized in terms of the growing rate of the domain and the degree of model misspecification/complexity.

Inference of Random Effects for Linear Mixed-Effects Models with a Fixed Number of Clusters

“…The coverage probabilities of our confidence interval tend to the nominal level (i.e., 0.95) as n increases for all cases even when m is very small. In contrast, the conventional method tends to be too optimistic for both σ 2 2 and σ 2 4 . For example, the coverage probabilities are less than 0.73 when m = 2 regardless of n. Although the coverage probabilities are a bit closer to the nominal level when m is larger, they are still in the range of (0.82, 0.87) when m = 10, showing that the conventional confidence interval is not valid for small m.…”

“…Theorem 1. Consider the data generated from (2) with the true parameters given by (6). Let (α, γ) ∈ A 0 × G 0 be a correct model defined in (4).…”

“…Note that for q = 1, Z i = z i,1 and b i = b i,1 . Note that by Lemma 5, we consider the sample space (σ 2 1 , v 2 ) ∈ (0, ∞) 2 . We first derive the explicit forms of the ML estimators θ1 and v2 .…”

“…This is because when m is fixed, the Fisher information for random-effects variances fails to grow with N , and hence the corresponding ML estimators do not achieve consistency. A similar difficulty arises in a spatial-regression model of Chang et al [2] under the fixed domain asymptotics, in which the spatial covariance parameters cannot be consistently estimated. A direct impact of this difficulty is that the classical central limit theorem established under m → ∞ for the ML (or REML) estimators [7,8,18] is no longer valid.…”

Mixed Domain Asymptotics for Geostatistical Processes

“…However, asymptotics of the mixed domain framework is underdeveloped for likelihood-based methods. Recently, Chang et al (2017) established consistency and derived the limit distribution of maximum likelihood estimators for an Ornstein-Uhlenbeck process under the mixed domain asymptotic framework. In this paper, we will focus on establishing some general conditions of mixed domain asymptotics for maximum likelihood estimators.…”