The Estimates Of Regression Coefficient Matrix And Covariance Matrices In The Growth Curve Model With Random Effects

The growth curve model is a wide-ranging linear model. Many discussions have been devoted by scholars to it in theoretical and applied aspects . A.P.Verbyla and W.N.Venables(1988) extended the ordinary growth curve model into the extension of growth curve model and obtained an estimate of the unknown parameter matrix under the conditions that the matrix of observations follow multivariate normal distribution,where every design matrix has full column rank[7].Anderson , Amemiva and Fujisawa et al scholars extended the growth curve model into the one with random effects and considered the likelihood ratio criterion (LRC) for its mean structure ,where the random vectors follow normal distribution on the assumptions that random effects and random errors are mutually independent[1][2] .In this thesis, we consider the extension of growth curve model: where are random matrices , are known design matrices , is unknown matrix , , mutually independent , are mutually independent, and are mutually independent, , (it is existent and finite), (it is existent and finite) . This paper is organized into three chapter. Firstly therelative backgrounds of this model and its recent research results are introduced in chapter 1 . In chapter 2 ,we obtain the best linear unbiased estimate of theestimable linear function (K L)B by vectoring KBL' ( in the means ofnonnegative definite ) with Albert method ,while this results form a weaker condition .its formula is:In chapter 3, least square estimates of covariance matrices ,r and their linear function tr(C + D) are obtained by using the projective theory and the spectrum decomposition of matrix. Finally, we give some optimal properties of tr(C + D) .The main results are as follow:tr(C∑*+ DГ*) is unbiased estimate of tr(C∑ + DГ) if and only if D = M2ZW2.