The final component of Var Y to be estimated is (E - MWE)' W (E - MWE) which 

 equals E'WE - E'WMWE, or returning to the summation notation: 



N N i-1 N N i-l 



Z s.E.^ + 2 Z Z s.E.E.s. - E s.2£'.2 - 2 Z T. s .E .E .s . 

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



Taking expectations, and noting that each element E^ is the average of n residuals 



2/1/ 2 ^ 2 ^ 



e (E-MWE)' W(E-MWE) = — E s . - — Z s .2 = H_ (i _ j; 5.2-, 



n -L n 1 n ^ 



i=l i=l i=l 



Putting all the components together, 



^ . 1 -2 ^ 



Var y = n2 b'Z'WZB - 52 tr [Z ' WZ( z ' z) ' -^l + ^ (i - i s.2) 



^=l 



in which 



52 = (-pY _ bz'r)/ (n - - 1) 



and 



tr(C) = Z C. 

 r=l 



When = 1 , the above matrix equation reduces to the variance estimate in the 

 previous section for the linear model. Obviously the order q of the polynomial is 

 not known when PPSORT is executed, so the qy.q matrix Z'WZ must be calculated 

 subsequently by a procedure analogous to the computation of TS in PPSORT. Such a 

 program could be prepared by omitting from PPSORT the selection of random sample 

 units and by making SX a t^-element vector of powers of the difference CP(J+1) - CP 



8 



