vartaylor_ratio {sampling}R Documentation

Taylor-series linearization variance estimation of a ratio

Description

Computes the Taylor-series linearization variance estimation of the ratio hat(Ys)/hat(Xs).

Usage

vartaylor_ratio(Ys,Xs,pikls)

Arguments

Ys vector of the first observed variable; its length is equal to n, the sample size.
Xs vector of the second observed variable; its length is equal to n, the sample size.
pikls matrix of the joint inclusion probabilities of the sample units; its dimension is nxn.

Details

The function implements the following estimator:

hat{Var}(frac{hat{Ys}}{hat{Xs}})=sum_{iin s}sum{jin s}frac{π_{ij}-π_iπ_j}{π_{ij}}frac{hat{z_i}hat{z_j}}{π_iπ_j}

where hat{z_i}=frac{(Ys_i-hat{r}Xs_i)}{hat{Xs}}, hat{r}=hat{Ys}/hat{Xs}, hat{Ys}=sum{Ys_i/π_i}, hat{Xs}=sum{Xs_i/π_i}.

References

Woodruff, R. (1971). A Simple Method for Approximating the Variance of a Complicated Estimate, Journal of the American Statistical Association, Vol. 66, No. 334 , pp. 411–414.

Examples

# Belgian municipalities data base
data(belgianmunicipalities)
attach(belgianmunicipalities)
# Computes the inclusion probabilities
pik=inclusionprobabilities(averageincome,100)
# the first variable on the population
Y=Men03
# the second variable on the population
X=Women03
# population size
N=length(pik)             
# matrix of joint inclusion prob. on the population for Tille sampling
pikl=UPtillepi2(pik)     
# draw a sample using the Tille sampling 
s=UPtille(pik)           
# inclusion probabilities on the sample
piks=pik[s==1]            
# the first observed variable
Ys=Y[s==1]
# the second observed variable
Xs=X[s==1]              
# matrix of joint inclusion prob. on the sample          
pikls=pikl[s==1,s==1] 
# computes the ratio HTestimator(Ys,piks)/HTestimator(Xs,piks) and its estimated variance
vartaylor_ratio(Ys,Xs,pikls)

[Package sampling version 2.3 Index]