The Minimum Norm Estimation Of Covariance Matrix In The Extended Growth Curve Model

This paper considers an extended growth curve model: where Y = (y(1),y(2},-~,y(n)) is .observable random matrix, are known and n s matrices respectively, Bi is an unknown &,. x/, matrix of regression coefficients ( i = 1,2,...m ) , is an s p random error matrix , are independent and (it is existent and finite),i= 1,...s , where :There exists a p-dimensional random vector such that . At the same time we suppose For any given matrix , wegive the MINQE(U,I) of tr(C∑) when it is estimable, and obtain the necessary and sufficient conditions for the MINQE(U,I) to be the UMVIQUE. At last ,we discuss the necessary and sufficient conditions for the UMVIQUE of tr(C∑) to exist. The main results are as follows:(l)The necessary and sufficient condition for tr(C∑) to be estimable is given .(2)The unique (a.s) MINQE(U,I) of tr(C∑) is y'MA*My ,where(3)In the case of i.i.d, the necessary and sufficient condition is obtained for MINQE(U,I) y'MA*My to be the UMVIQUE of tr(C) by and for for some , and (4)The necessary and sufficient condition for the UMVIQUE of tr(C∑) to exist is the necessary and sufficient condition for the MINQE(U,I) of tr(C∑) to be the UMVIQUE of tr(C∑).