The Best Linear Unbiased Estimate Of Parameter Matrix And Measures Of Inefficiency Of The Least Squares Estimate In Singular Growth Curve Model

Consider singular growth curve model as follows : Y = ABC + E , where A , C are all known matrices, B is an unknown matrix of regression coefficients , Y is a matrix of observations and E = ( (1), (2) ,... (n))' is matrix of random errors . We make the following assumption for When 2 is positive definite matrix, different estimators about matrix of regression coefficients and inefficiency of Least squares estimate have been discussed in many documents. Considered 2 is nonnegative definite matrix, this thesis derives Best linear unbiased Estimate of parameter matrix B and estimable parameter function KBL under the meaning of matrix nonnegative definite and the property of maximum probability of BLUE is investigated. next, we discuss some necessary and sufficient conditions of the equality of the LSE and BLUE, then we derive the estimation of the deviation bet-ween the Least squares and the Best Linear unbias Estimators of the mean matrix, meanwhile a relative efficiency of LSE ofB is proposed and its bound is given . the main results are as follows:(1) Under the meaning of matrix nonnegative definite, the Best Linear un-biased Estimate of estimate function KBL is KB * L = KA + Y [ I - M( M M) + M ]C+L.and if B is estimable, then BLUE of B is B * = A + Y[ I - M( M E M) + M(2) T/ze L珃s?squares Estimate of ABC equals to the Best Linear Unbiased estimate if and only if one of the following Conditions holds : i )(I-C+C)Q+ VC+C = 0(3) The estimation of the deviation bet-ween the Least squares and the Best Linear Unbiased Estimators of the mean matrix is as folio-wing :(5 ) If vec ( E ) ~ EC ( 0 , I , ) , then for arbitrary linear unbiased estimate KBL of KBL and nonnegati-ve definite matrix Sn n and real unmber c we haveP{(vec(KB* L-KBL)YS(vec(KB* L~KBL) P{ ( vec(KBL - KBL )YS(vec(KBL - KBL ) ) c2 } .