With this notation, define three additional matrices: 



Z = X - MWX of dimension Nxq 



B = (Z'WZ)"-^ Z'WR " " qxl 



E = R - ZB I. .1 717x1 



in which Z' is the transpose of Z. 



Now Z'WZ is the analog of 



717 _ _ N 



n2 z (x^ - x)2 / ( E a:^ )2 



k=l i=l 

 and (R-MWR) 'WCR-MWR) is the analog of 

 i7 _ _ E 



k=l i=l 



in Hartley's notation. 



Furthermore, because MWM = M, it can be shown that MWZ = 0, the null matrix of 



dimension nxq. It is also true that Z'WE = 0, the null vector of dimension 

 qxl . 



The objective is to derive an estimate for the variance of the estimate of the 



population total Y = '^''"|^- The variance of Y is given by 



Var J = n2 (R-MWR) ' W (R-MWR) 



= n2 (ZB + E - MWZB - MWE)' W (ZB+E - MWZB - MWE) 

 = n2 (ZB + E - MWE)' W (ZB+E - MWE) 



= n2 (B'Z'WZB + (E - MWE)' W (E - MWE) + 2 B'Z'W (E-MWE)) 

 The last product is zero because 

 B'Z'WE = B'O = 



and 



B'Z'WMWE = B'(MWZ)' WE = B'O'WE = 



In the variance expression, the first matrix product, Z'WZ is the multivariate analog 

 of the factor TS computed in PPSORT. 



To estimate the B'Z'WZB component of Var Y requires first an estimate of B based 

 only on the sample. Let b represent the vector of q estimates of the elements of B, 

 and let z be the matrix of deviations of the q powers of for each of the n units 

 in the sample: 



n n . n 



Z = (P. - E P./n, P ^ - I P ."^ fn, . . . , P - 1 p3/n) 

 i=l ^ «^ i=l ^ ^ i=l ^ 



for J = 1, . . . , n. 



6 



