sampling (version 0.45)

UPtille: Till� sampling

Description

Uses the Till� method to select a sample of units (unequal probabilities, without replacement, fixed sample size).

Usage

UPtille(pik,eps=1e-6)

Arguments

pik

the vector of the prescribed inclusion probabilities.

eps

the control value, by default equal to 1e-6.

encoding

latin1

Details

Returns a vector (with elements 0 and 1) of size N, the population size. Each element k of this vector indicates the status of unit k (1, unit k is selected in the sample; 0, otherwise). The value eps is used to control pik (pik>eps & pik < 1-eps).

References

Till�, Y. (1996), An elimination procedure of unequal probability sampling without replacement, Biometrika, 83:238-241. Deville, J.-C. and Till�, Y. (1998), Unequal probability sampling without replacement through a splitting method, Biometrika, 85:89-101.

See Also

UPsystematic

Examples

############
## Example 1
############
#define the prescribed inclusion probabilities
pik=c(0.2,0.7,0.8,0.5,0.4,0.4)
#select a sample
s=UPtille(pik)
#the sample is
(1:length(pik))[s==1]
############
## Example 2
############
# Selection of samples of municipalities       
# with equal or unequal probabilities.         
# Comparison of the accuracy by boxplots.                                                 
b=data(belgianmunicipalities)
pik=inclusionprobabilities(belgianmunicipalities$Tot04,200)
N=length(pik)
n=sum(pik)
#number of simulations; for an accurate result, increase this value 
sim=10
ss=array(0,c(sim,8))
# the interest variable
y=belgianmunicipalities$TaxableIncome
# simulation and computation of the Horvitz-Thompson estimator
for(i in 1:sim)
{
cat("Step ",i,"")
ss[i,]=ss[i,]+c(
HTestimator(y,pik,UPpoisson(pik)),
HTestimator(y,pik,UPrandomsystematic(pik)),
HTestimator(y,pik,UPrandompivotal(pik)),
HTestimator(y,pik,UPtille(pik)),
HTestimator(y,pik,UPmidzuno(pik)),
HTestimator(y,pik,UPsystematic(pik)),
HTestimator(y,pik,UPpivotal(pik)),
HTestimator(y,rep(n/N,N),srswor(n,N)))
}
# boxplots of the estimators
colnames(ss) <- 
c("poisson","rsyst","rpivotal","tille","midzuno","syst","pivotal","srswor")
boxplot(data.frame(ss), las=3)
# The results  of the simulations can be interpreted.
# Simple random sampling 
# and Poisson sampling are not accurate.
# All unequal probability sampling methods seem
# to have the same accuracy, except systematic sampling and pivotal sampling 
# that have variances which depend on the order of the units in the file.