## Estimation in a growth curve model with singular covariance.(English)Zbl 1015.62056

Summary: Let $$Y$$ be a multivariate normal random matrix with covariance $$A\otimes\Sigma$$ and mean $$\mu\in S_1S_2'$$, where $$S_i=\{X_i b_i: K_i'b_i= M_i'u_i$$ for some $$u_i\}$$ and $$S_1S_2'$$ is the linear span of the set of all $$x_1x_2'$$ with $$x_i\in S_i$$. Explicit formulae are obtained for the estimators of $$(\mu,\Sigma)$$. These estimators are investigated through a large class of loss functions and other principles. None of the matrices $$A,\Sigma, X_i,K_i$$ and $$M_i$$ are assumed to have full column rank. For robust studies, elliptical $$Y$$ is considered when it is appropriate.

### MSC:

 62H12 Estimation in multivariate analysis 62J05 Linear regression; mixed models
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### References:

 [1] Anderson, T.W., An introduction to multivariate analysis., (1984), Wiley New York · Zbl 0651.62041 [2] Anderson, T.W.; Olkin, I., Maximum-likelihood estimation of parameters of a multivariate normal distribution, Linear alg. appl., 70, 147-171, (1985) · Zbl 0586.62074 [3] Arnold, S.F., The theory of linear models and multivariate analysis., (1981), Wiley New York [4] Calvert, B.; Seber, G.A.F., Minimization of functions of a positive semidefinite matrix A subject to AX=0, J. multivariate anal., 8, 173-180, (1978) [5] Chan, N.N.; Keung, T.K., Linear hypotheses made explicit, Commun. statist. theor.-meth., 16, 387-400, (1997) · Zbl 0898.62085 [6] Dykstra, R.L., Establishing the positive definiteness of the sample covariance matrix, Ann. math. statist., 41, 2153-2154, (1970) · Zbl 0212.22202 [7] Eaton, M.L., Multivariate statistics., (1983), Wiley New York · Zbl 0587.62097 [8] Fujikoshi, Y., On the asymptotic non-null distributions of the LR criterion in a general MANOVA, Canadian J. statist., 2, 1-12, (1974) · Zbl 0374.62066 [9] Gupta, A.K.; Varga, T., Elliptically contoured models in statistics., (1993), Kluwer Academic Publishers Boston · Zbl 0789.62037 [10] Kruskal, W., The geometry of generalized inverse, J. roy. statist. soc. ser., B 37, 272-283, (1975) · Zbl 0318.15003 [11] Muirhead, R.J., Aspects of multivariate statistical theory., (1982), Wiley New York · Zbl 0556.62028 [12] Potthoff, R.F.; Roy, S.N., A generalized multivariate analysis of variance model useful especially for growth curve problems, Biometrika, 51, 313-326, (1964) · Zbl 0138.14306 [13] Rao, C.R., Linear statistical inference and its applications., (1973), Wiley New York · Zbl 0169.21302 [14] Rao, C.R.; Mitra, S.K., Generalized inverse of matrices and its applications., (1971), Wiley New York [15] von Rosen, D., The growth curve model: a review, Commun. statist.-theor. meth., 20, 9, 2791-2822, (1991) · Zbl 0800.62450 [16] von Rosen, D., Moments of estimators, Statistics, 22, 111-131, (1991) · Zbl 0738.62067 [17] Wong Chi Song, On the use of differentials in statistics, Linear algebra appl., 70, 282-299, (1985) · Zbl 0592.62058 [18] Wong Chi Song, Modern analysis and algebra., (1986), Xian University Press Xian [19] Wong Chi Song, Linear models in a general parametric form, Commun. statist.-theory meth., 18, 8, 3095-3115, (1989) · Zbl 0696.62281 [20] Wong Chi Song, Linear models in a general parametric form, Sankhya, ser., A 55, 130-149, (1993) · Zbl 0791.62075 [21] Wong Chi Song, Liu Dongsheng, 1995. Moments of generalized Wishart distributions. J. Multivariate Anal. 52 280-294. · Zbl 0877.62051 [22] Wong Chi Song, Masaro, Joe, Deng, Weicai, 1995. Estimating covariance in a growth curve model. Linear Algebra Appl. 214, 103-118. · Zbl 0812.62062 [23] Wong Chi Song, Masaro, Joe, Wang, T., 1991. Multivariate versions of Cochran theorems. J. Multivariate Anal. 39, 154-174. · Zbl 0749.62037
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