Bernoulli

  • Bernoulli
  • Volume 14, Number 3 (2008), 623-636.

Asymptotic normality and consistency of a two-stage generalized least squares estimator in the growth curve model

Jianhua Hu and Guohua Yan

Full-text: Open access

Abstract

Let $\mathbf{Y}=\mathbf{X \Theta Z}'+\mathbf{\mathcal {E}}$ be the growth curve model with $\mathbf{\mathcal{E}}$ distributed with mean $\mathbf{0}$ and covariance $\mathbf{I}_n⊗\mathbf{Σ}$, where $\mathbf{Θ}$, $\mathbf{Σ}$ are unknown matrices of parameters and $\mathbf{X}$, $\mathbf{Z}$ are known matrices. For the estimable parametric transformation of the form $\mathbf{γ}=\mathbf{CΘD}'$ with given $\mathbf{C}$ and $\mathbf{D}$, the two-stage generalized least-squares estimator $\mathbf{γ̂}(\mathbf{Y})$ defined in (7) converges in probability to $\mathbf{γ}$ as the sample size $n$ tends to infinity and, further, $\sqrt{n}[\hat{\mathbf{\gamma}}(\mathbf{Y})-{\mathbf{\gamma}}]$ converges in distribution to the multivariate normal distribution $\mathcal{N}(\mathbf{0},(\mathbf{C}\mathbf{R}^{-1}\mathbf{C}')\otimes(\mathbf{D}(\mathbf{Z}'\mathbf{\Sigma }^{-1}\mathbf{Z})^{-1}\mathbf{D}'))$ under the condition that $\lim_{n→∞} \mathbf{X}'\mathbf{X}/n=\mathbf{R}$ for some positive definite matrix $\mathbf{R}$. Moreover, the unbiased and invariant quadratic estimator $\mathbf{\hat Σ}(\mathbf{Y})$ defined in (6) is also proved to be consistent with the second-order parameter matrix $\mathbf{Σ}$.

Article information

Source
Bernoulli, Volume 14, Number 3 (2008), 623-636.

Dates
First available in Project Euclid: 25 August 2008

Permanent link to this document
https://projecteuclid.org/euclid.bj/1219669623

Digital Object Identifier
doi:10.3150/08-BEJ128

Mathematical Reviews number (MathSciNet)
MR2537805

Zentralblatt MATH identifier
1155.62014

Keywords
asymptotic normality consistent estimator estimation generalized least-squares estimator growth curve model

Citation

Hu, Jianhua; Yan, Guohua. Asymptotic normality and consistency of a two-stage generalized least squares estimator in the growth curve model. Bernoulli 14 (2008), no. 3, 623--636. doi:10.3150/08-BEJ128. https://projecteuclid.org/euclid.bj/1219669623


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