Derivation of Variance Estimate for Polynomial Model 



The variance estimate given above assumes that the trend of Y ./P. is a linear 

 function of P . . If the relation is known to be curvilinear in advance, a stiitable 

 transformation of the size variable should restore the linearity. However, if the 

 curvilinear trend is not observed until after the sample has been measured, the 

 variance estimate based on the linear model would be too large. In such a case, 

 the following analysis shows how a multivariate analog of the linear model could be 

 used to estimate the variance of the population total. 



Let the following symbols represent vectors or matrices of the indicated 

 dimensions. The subscript k designates the ^:th sample cluster defined by the sorted 

 order and the sampling interval. The range of k is from 1 to 



r, = (Y./P.) of dimension nxl 



m = (1) " " nxl 



R = (m'r|^) " " iVxl 



W = Cs^6^p " W 



• M = (1) " " /l^xiV 



In the above vectors and matrices, the scalars are defined as follows: 



Y^ is the measured value of the variable of interest for the ith unit in 



the population. 



P. is the probability with which Y . is measured. 



Sj, is the probability that the ?cth sample olustev of Y. is measured. 



1 if i = J 

 if -i / J 



N is the total number of units in the population. 



n is the number of units in the sample. 



q is the degree of the polynomial function of P^. 



Note that M is a matrix and m is a vector having unity in all elements. The effect of 

 multiplying a vector by M or m is to replace each element in the vector with the 

 column total. In particular, 



N 



MWM = M because z's, = 1 



6 . . = 



5 



