## Uniformly minimum variance nonnegative quadratic unbiased estimation in a generalized growth curve model.(English)Zbl 1157.62035

Summary: Consider the generalized growth curve model $$Y=\sum_{i=1}^m X_iB_iZ_i^\prime +U\mathcal E$$ subject to $$R(X_m)\subseteq \cdots \subseteq R(X_{1})$$, where $$B_i$$ are the matrices of unknown regression coefficients, and $$\mathcal E=(\varepsilon_1,\dots ,\varepsilon_s)^\prime$$ and $$\varepsilon_j$$ $$(j=1,\dots,s)$$ are independent and identically distributed with the same first four moments as a random vector normally distributed with mean zero and covariance matrix $$\Sigma$$. We derive necessary and sufficient conditions under which the uniformly minimum variance nonnegative quadratic unbiased estimator (UMVNNQUE) of the parametric function tr$$(C\varSigma)$$ with $$C\geq 0$$ exists. Necessary and sufficient conditions for a nonnegative quadratic unbiased estimator $$\mathbf y^\prime A\mathbf y$$ with $$\mathbf y = \text{Vec}(Y^\prime)$$ of tr$$(C\Sigma)$$ to be the UMVNNQUE are obtained as well.

### MSC:

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

### Keywords:

generalized growth curve model; UMVNNQUE
Full Text:

### References:

 [1] 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 [2] Rosen, D.V., The growth curve model: A review, Communication in statistics theory and methods, 20, 2791-2822, (1991) · Zbl 0800.62450 [3] Kshirsagar, A.M.; Smith, W.B., Growth curves, (1995), Marcel Dekker New York [4] Pan, J.X.; Fang, K.T., Growth curve models and statistical diagnostics, (2002), Springer-Verlag New York · Zbl 1024.62025 [5] D.V. Rosen, Maximum likelihood estimates in multivariate linear normal models with special references to the growth curve model, Research Report 135, Dept. of Math. Statist., University of Stockholm, Stockholm, Sweden, 1984 [6] Verbyla, A.P.; Venables, W.N., An extension of the growth curve model, Biometrika, 75, 129-138, (1988) · Zbl 0636.62073 [7] Evans, J.C.; Roberts, E.A., Analysis of sequential observations with applications to experiments on grazing animals and perennial plants, Biometrics, 35, 687-693, (1979) [8] Rosen, D.V., Maximum likelihood estimation in multivariate linear normal models, Journal of multivariate analysis, 31, 187-200, (1989) · Zbl 0686.62037 [9] Hsu, P.L., On the best unbiased estimate of variance, Statistical research mem., 2, 91-104, (1938) [10] Kleffe, J., On hsu’s theorem in multivariate regression, Journal of multivariate analysis, 9, 442-452, (1979) · Zbl 0435.62052 [11] Yang, W.L., $$M I N Q E(U, I)$$ and $$U M V I Q U E$$ of the covariance matrix in the growth curve model, Statistics, 26, 49-59, (1995) [12] Wu, X.Y.; Zou, G.H.; Chen, J.W., Unbiased invariant minimum norm estimation in generalized growth curve model, Journal of multivariate analysis, 97, 1718-1741, (2006) · Zbl 1112.62054 [13] X.Y. Wu, H. Liang, G.H. Zou, The least square estimation in Generalized Growth Curve Model, Technical Report, Department of Biostatistics and Computational Biology, University of Rochester, 2007 [14] Hamid, J.S.; Rosen, D.V., Residuals in the extended growth curve model, Scandinavian journal of statistics, 33, 121-138, (2006) · Zbl 1125.62054 [15] Jiang, W.J., The best nonnegative estimation of covariance matrix in the growth curve model, Science in China (series A), 12, 1253-1263, (1992) [16] Kelly, R.J.; Mathew, T., Improved nonnegative estimation of variance components in some mixed models with unbalanced data, Technometrics, 36, 171-181, (1994) · Zbl 0925.62096 [17] Mathew, T.; Niyogi, A.; Sinha, B.K., Improved nonnegative estimation of variance components in balanced multivariate mixed models, Journal of multivariate analysis, 51, 83-101, (1994) · Zbl 0806.62057 [18] Srivastava, M.S.; Kubokawa, T., Improved nonnegative estimation of multivariate components of variance, Annals of statistics, 27, 2008-2032, (1999) · Zbl 0961.62054 [19] Kubokawa, T.; Saleh, A.K.M.E.; Konno, Y., Bayes, minimax and nonnegative estimators of variance components under kullback – leibler loss, Journal of statistical planning and inference, 86, 201-214, (2000) · Zbl 0953.62066 [20] Sun, Y.J.; Sinha, B.K.; Rosen, D.V.; Meng, Q.Y., Nonnegative estimation of variance components in multivariate unbalanced mixed linear models with two variance components, Journal of statistical planning and inference, 115, 215-234, (2003) · Zbl 1041.62045 [21] You, J.H.; Sun, X.Q., The conditions of optimal nonnegative quadratic estimation in GMANOVA-MANOVA model, Journal of guyuan teachers college, 18, 1-5, (1997) [22] Rao, C.R.; Kleffe, J., Estimation of variance components and applications, (1988), North-Holland New York · Zbl 0645.62073 [23] Drygas, H.; Hupet, G., A new proof of hsu’s theorem in regression analysis—A coordinate-free approach, Mathematische operationsforschung und statistik, 8, 333-335, (1977) · Zbl 0374.62062
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.