## 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:

 [1] Bischoff, W. (2000). Asymptotical optimal tests for some growth curve models under non-normal error structure., Metrika 50 195-203. · Zbl 0992.62051 [2] 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 [3] 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 [4] 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 [5] Hu, J. (2008). Wishartness and independence of matrix quadratic forms in a normal random matrix., J. Multivariate Anal. 99 555-571. · Zbl 1132.62036 [6] 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 [7] Lehmann, E.L. (1999)., Elements of Large-Sample Theory . New York: Springer. · Zbl 0914.62001 [8] Lehmann, E.L. and Romano, J.P. (2005)., Testing Statistical Hypotheses , 3rd ed. New York: Springer. · Zbl 1076.62018 [9] Muirhead, R.J. (1982)., Aspects of Multivariate Statistical Theory . New York: Wiley. · Zbl 0556.62028 [10] Nussbaum, M. (1977). Asymptotic efficiency of estimators in the multivariate linear model., Math. Oper. Statist. Ser. Statist. 8 439-445. · Zbl 0392.62035 [11] 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 [12] Rao, C.R. (1973)., Linear Statistical Inference and Its Applications , 2nd ed. New York: Wiley. · Zbl 0256.62002 [13] Theil, H. (1971)., Principles of Econometrics . New York: Wiley. · Zbl 0221.62002 [14] Žežula, I. (1993). Covariance components estimation in the growth curve model., Statistics 24 320-330. · Zbl 0808.62052 [15] Žežula, I. (1997). Asymptotic properties of the growth curve model with covariance components., Appl. Math. 42 57-69. · Zbl 0898.62067 [16] Žežula, I. (2006). Special variance structures in the growth curve model., J. Multivariate Anal. 97 606-618. · Zbl 1101.62042
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