For general linear model : y=XB+u , u~N(0,(ssq)I) , u are i.i.d
Let b* be the OLS estimator of B ,ie. b*=inv(X'X)X'y .
Let b be any linear unbiased estimator of B having the form : b=Hy where H doesn't depend on y .
We want to see which one is more effective , b or b* :
By unbiasedness , E[b]=E[HXB+Hu]=HXB=B , which implies HX=I
Define C by H=inv(X'X)X'+C , then HX=I implies CX=0 . Thus b=Hy=[inv(X'X)X'+C]y=b*+Cy=b*+Cu (it can be shown that Cy=Cu)
To see which one is more effective , Cov(b)=Cov(b*)+ ssqCC' which gives Cov(b)-Cov(b*)=ssqCC' . The difference being p.s.d (thus b* is more effective estimator).
My question is that how can CC' be p.s.d ? I tried to use CX=0 but it looks very tricky becuause neither C nor X has to be 0 , could someone help ?
$CC'$ is positive semidefinite because we can write
$$x'CC'x=\|C'x\|^2 \ge 0$$
We do not require $CX=0$.