vartaylor_ratio {sampling} | R Documentation |
Computes the Taylor-series linearization variance estimation of the ratio hat(Ys)/hat(Xs).
vartaylor_ratio(Ys,Xs,pikls)
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. |
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}.
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.
# 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)