Calibration estimator and its variance estimation
Computes the calibration estimator of the population total and its variance estimation using the residuals' method.
calibev(Ys,Xs,total,pikl,d,g,q=rep(1,length(d)),with=FALSE,EPS=1e-6)
Ys |
vector of interest variable; its size is n, the sample size. |
Xs |
matrix of sample calibration variables. |
total |
vector of population totals for calibration. |
pikl |
matrix of joint inclusion probabilities of the sample units. |
d |
vector of initial weights of the sample units. |
g |
vector of g-weights; its size is n, the sample size. |
q |
vector of positive values accounting for heteroscedasticity; its size is n, the sample size. |
with |
if TRUE, the variance estimation takes into account the initial weights d; otherwise, the final weights w=g*d are taken into account; by default, its value is FALSE. |
EPS |
the tolerance in checking the calibration; by default, its value is 1e-6. |
If with is TRUE, the following formula is used
\hat{Var}(\hat{Ys})=∑_{k\in s}∑_{\ell\in s}((π_{k\ell}-π_kπ_{\ell})/π_{k\ell})(d_ke_k)(d_\ell e_\ell)
else
\hat{Var}(\hat{Ys})=∑_{k\in s}∑_{\ell\in s}((π_{k\ell}-π_kπ_{\ell})/π_{k\ell})(w_ke_k)(w_\ell e_\ell)
where e_k denotes the residual of unit k.
The function returns two values:
cest |
the calibration estimator, |
evar |
its estimated variance. |
Deville, J.-C. and Särndal, C.-E. (1992). Calibration estimators in survey sampling. Journal of the American Statistical Association, 87:376–382.
Deville, J.-C., Särndal, C.-E., and Sautory, O. (1993). Generalized raking procedure in survey sampling. Journal of the American Statistical Association, 88:1013–1020.
############ ## Example ############ # Example of g-weights (linear, raking, truncated, logit), # with the data of Belgian municipalities as population. # Firstly, a sample is selected by means of systematic sampling. # Secondly, the g-weights are calculated. data(belgianmunicipalities) attach(belgianmunicipalities) # matrix of calibration variables for the population X=cbind( Men03/mean(Men03), Women03/mean(Women03), Diffmen, Diffwom, TaxableIncome/mean(TaxableIncome), Totaltaxation/mean(Totaltaxation), averageincome/mean(averageincome), medianincome/mean(medianincome)) # selection of a sample of size 200 # using systematic sampling # the inclusion probabilities are proportional to the average income pik=inclusionprobabilities(averageincome,200) N=length(pik) # population size s=UPsystematic(pik) # draws a sample s using systematic sampling Xs=X[s==1,] # matrix of sample calibration variables piks=pik[s==1] # sample inclusion probabilities n=length(piks) # sample size # vector of population totals of the calibration variables total=c(t(rep(1,times=N))%*%X) g1=calib(Xs,d=1/piks,total,method="linear") # computes the g-weights pikl=UPsystematicpi2(pik) # computes the matrix of the joint inclusion probabilities pikls=pikl[s==1,s==1] # the same matrix for the units in s Ys=Tot04[s==1] # the variable of interest is Tot04 (for the units in s) calibev(Ys,Xs,total,pikls,d=1/piks,g1,with=FALSE,EPS=1e-6)
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