The Estimate Of The Regression Parameter Matrix With Some Restrictions In Growth Curve Model

Consider growth curve model as follows:E(Y_n×p)=X_n×kB_k×r Z_r×pwhere X , Z are known design matrix ,B is an unknown matrix of regression coefficient , kn×p are independent p-variate vectors with common covariance matrix s²I_p ands² is unknown .Different estimators of regression coefficients B have been discussed in many documents . When regression coefficients B is with some restrictions ,forexample , GBH = D or GB = F₁ BH = F₂ , in general,the idea is to find thesolution of the equations and then to replace the regression coefficients B with the solution,which reduces the restricted GCM to an unrestricted GCM. But the resulting expressions of the estimates are somewhat complicated .this thesis finds some conditions to make the estimators simple .the main results are as follows: (1) When the regression coefficients matrix B is subject to the restrictions:GB = 0,BH = 0a) if μ(G')∩μ(X') = {0} and μ{H)∩μ{Z) = {Q} then for an estimablelinear function KBL , the LSE KBL is a best linear unbiased estimate (in the sense of nonngative definite).b) if μ(G')∩μ(X') = {O},μ(H)∩μ(Z) = {O},R=k,R(Z:H)=r thenB₂* = (X'X + G'G)^-1 X'YZ (ZZ' + HH' )^-1 is a best linear unbiased estimate of B. (2) When the regression coefficients matrix B is subject to the restrictions:GBH = 0 if μ(G')∩μ(X')={0} or μ(H)∩μ(Z)= {0} then for an estimablelinear function KBL , the LSE KBL is a best linear unbiased estimate (in the sense of nonngative definite).