The Relative Efficiency Of The Parameter Estimate In Singular Gauss-Markoff Model

Consider the model of Gauss-Markoff Y=Xβ+ε E(ε)=0, cov(ε)=σ~2, (1)where Y is n×1 observed date vector,X is n×p design matrix, it's rank r(X)≤p , β is p×1 unknown parameter, ε is n×1 random error vector, σ~2is unknown parameter , ∑ is n×n covariance matrix. When r=p,∑ >0, some kinds of relaive efficiency have been introduced to meassure parameter beta's least square estimator (LSE) compare to the best linear unbiased estimator (BLUE) at the researsh documentation [3],[4] ,at the same time,the relation between the lower bound and generalized relative coefficient has also built.When r0. but βs LSE β|^=(XX)~+XY. .it's covariance matrix is cov(β|^)=(XX)~+X∑(XX) σ~2Based on the theorem of Gauss-Markoff,we know cov(β|^)≥cov(β~+), we usually find the case that is unknown or not clear enough in practical application,so we substite the β|^ by β~+, but that can bring lost to accuracy of estimation.the relative efficiency is an important method to meassure the lost which created by this substition.In this text ,two new relative efficiency (e() ) are given and the lower bound of them are studied,fuethermore,the eficiency defined by using Euclid norm from WangJie is popularized and it's bound is studied.The three new relative efficiency are as fowong:e( )= , e ()=, e(= where ,>0 are the characteristic roots of cov() and cov() respectively, e( is extended from the relative efficiency defined by WangJie in researh documentation [3]。=[tr(AA)]. Furthermore,the relation between relative efficiency and generalized coefficient is also built,of course, the reasonability and advantage or disadvantage of this three relative efficiency are discussed in the text.Preliminary knowedge and main results are as following:Lemma2.1 Let A,B is n real symmetry matrix,,and B0 ,we have(B)(A)(ABA)(B)(A) i=1,2, nLemma2.2 (Neumann inequality)Let A,B is n Hermite matrix,,>0 are their characteristic roots respectively,then trABLemma 2.3 (Poincare seprate theorem)Let A is n n Hermite matrix,U is nk column orthogonal matrix,that is U^{'}U=I_k ,then we have i=1,2,,kTheorem3.1 under the model of (1),we have eTheorem3.2 under the model of (1), we have eTheorem3.3 under the model of (1),for s any two unbiased estimation ,if cov()cov()0, then e() e(), i=1,2. Theorem 4.1 under the model of (1),for s any two unbiased estimation ,we have e(r Theorem4.2 under the model of (1), for s any two unbiased estimation ,if cov()cov()0, then e () e (), i=1,2. Theorem6.2 under the model of (1),if X is column orthogonal matrix and >0 ,we have eTheorem6.3 under the model of (1),if r(X)=p, r()=k p, then ewhere m=r(T) Theorem 6.5 under the model of (1),if r(X)=p, r()=k p, then ewhere m=r(T)...