## Asymptotic normality and consistency of a two-stage generalized least squares estimator in the growth curve model.(English)Zbl 1155.62014

Summary: Let $$Y = X\Theta Z' + {\mathcal E}$$ be a growth curve model with $$\mathcal E$$ distributed with mean 0 and covariance $$I_n\otimes \Sigma$$, where $$\Theta,\Sigma$$ are unknown matrices of the parameters and $$X,\;Z$$ are known matrices. For the estimable parametric transformation of the form $$\gamma = C\Theta D'$$ with given $$C$$ and $$D$$, the two-stage generalized least-squares estimator $\widehat\gamma(Y)=C(X'X)^{-1}X'Y\widehat\Sigma^{-1}(Y)Z(Z'\widehat\Sigma^{-1}(Y)Z)^{-1}D'$
converges in probability to $$\gamma$$ as the sample size $$n$$ tends to infinity and, further, $$\sqrt{n}[\widehat\gamma(Y)-\gamma]$$ converges in distribution to the multivariate normal distribution ${\mathcal N}(0, (CR^{-1} C')\otimes (D(Z'\Sigma^{-1} Z)^{-1}D'))$ under the condition that $$\lim_{n\to\infty} X'X/n=R$$ for some positive definite matrix $$R$$. Moreover, the unbiased and invariant quadratic estimator
$\widehat\Sigma(Y)=Y'WY, \quad W\equiv (n-\operatorname{rank} X)^{-1}(I-P_X),\quad P_X=X(X'X)^-X',$
is also proved to be consistent with the second-order parameter matrix $$\Sigma$$.

### MSC:

 62F12 Asymptotic properties of parametric estimators 62H12 Estimation in multivariate analysis

### Keywords:

generalized least-squares estimator
Full Text:

### References:

  Bischoff, W. (2000). Asymptotical optimal tests for some growth curve models under non-normal error structure., Metrika 50 195-203. · Zbl 0992.62051  Chakravorti, S.R. (1976). On asymptotic properties of the maximum likelihood estimates of the general growth curve model., Ann. Inst. Statist. Math. 28 349-357. · Zbl 0367.62083  Eicker, F. (1963). Asymptotic normality and consistency of the least squares estimators for families of linear regressions., Ann. Math. Statist. 34 447-456. · Zbl 0111.34003  Gong, L. (1998). Asymptotic distribution of likelihood ratio statistic for testing sphericity in the growth curve model., Acta Math. Sci. Ser. B Engl. Ed. 18 440-448. · Zbl 0916.62015  Hu, J. (2008). Wishartness and independence of matrix quadratic forms in a normal random matrix., J. Multivariate Anal. 99 555-571. · Zbl 1132.62036  Hu, J. and Shi, S. (2008). On the expressions of estimability in testing general linear hypotheses., Comm. Statist. Theory Methods 37 782-790. · Zbl 1318.62190  Lehmann, E.L. (1999)., Elements of Large-Sample Theory . New York: Springer. · Zbl 0914.62001  Lehmann, E.L. and Romano, J.P. (2005)., Testing Statistical Hypotheses , 3rd ed. New York: Springer. · Zbl 1076.62018  Muirhead, R.J. (1982)., Aspects of Multivariate Statistical Theory . New York: Wiley. · Zbl 0556.62028  Nussbaum, M. (1977). Asymptotic efficiency of estimators in the multivariate linear model., Math. Oper. Statist. Ser. Statist. 8 439-445. · Zbl 0392.62035  Potthoff, R.F. and Roy, S.N. (1964). A generalized multivariate analysis of variance model useful especially for growth curve problems., Biometrika 51 313-326. JSTOR: · Zbl 0138.14306  Rao, C.R. (1973)., Linear Statistical Inference and Its Applications , 2nd ed. New York: Wiley. · Zbl 0256.62002  Theil, H. (1971)., Principles of Econometrics . New York: Wiley. · Zbl 0221.62002  Žežula, I. (1993). Covariance components estimation in the growth curve model., Statistics 24 320-330. · Zbl 0808.62052  Žežula, I. (1997). Asymptotic properties of the growth curve model with covariance components., Appl. Math. 42 57-69. · Zbl 0898.62067  Žežula, I. (2006). Special variance structures in the growth curve model., J. Multivariate Anal. 97 606-618. · Zbl 1101.62042
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.