Finally, let r be the n element column vector of ratios: 



Then define the estimate of B to be 



b = (z'z)"^ z'r 



Next, define Z as the expected value of 



(b-B) 'Z'WZ(b-B) 



Because Z'WZ is a matrix of constants, 



Z = e (b'Z'WZb) - B'Z'WZB 



in which e (•) means "the expected value of" (•)• 



Under the assumption that r j.s drawn from a superpopulation having a mean of zB 

 and variance equal to e (e'e) = o^, 



(b-B) = (z'z)"^ z' (zB+e) - B 



= B + (z'z)"^ z'e - B 



= (z'z)"^ z'e 



Then, 



e (b-B)' Z'WZ (b-B) = e (e'z(z'z)"^ Z'WZ(z'z)"^ z'e) 



To simplify notation, designate the symmetric matrix between e' and e by C , then 



n n i-1 



e(e'Ce) = Z e (e C. + 2 H Z e (e.e.C.) 

 i=l ^ i=l j=l ^ ^ 



The expected value of is and the expected value of ^^^j is zero if the errors 

 are independent for a sufficiently high degree polynomial. 



Therefore, 



n 



Z = a2 Z C .. = tr(C) 

 ^=l 



Because one can commute matrices under the trace operator, 

 tr(C) = tr (z(z'z)"^ Z'WZ (z'z)"^ z') 

 = tr (Z'WZ (z'z)"^ z'z (z'z)"^ 

 = tr (Z'WZ (z'z)"^) 



Hence, Z can be estimated by the residual mean square error times the sum of the 

 diagonal elements of Z'WZ(z'z)~l. 



