Taylor-series linearization variance estimation of a ratio
Computes the Taylor-series linearization variance estimation of the ratio
\frac{\widehat{Y}_s}{\widehat{X}_s}.
The estimators in the ratio are Horvitz-Thompson estimators.
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 joint inclusion probabilities of the sample units; its dimension is nxn. |
The function implements the following estimator:
\widehat{Var}(\frac{\widehat{Ys}}{\widehat{Xs}})=∑_{i\in s}∑_{j\in s}\frac{π_{ij}-π_iπ_j}{π_{ij}}\frac{\widehat{z_i}\widehat{z_j}}{π_iπ_j}
where \widehat{z_i}=(Ys_i-\widehat{r}Xs_i)/\widehat{X}_s, \widehat{r}=\widehat{Y}_s/\widehat{X}_s, \widehat{Y}_s=∑_{i\in s}{Ys_i/π_i}, \widehat{X}_s=∑_{i\in s}{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.
data(belgianmunicipalities) attach(belgianmunicipalities) # inclusion probabilities, sample size 200 pik=inclusionprobabilities(Tot04,200) # the first variable (population level) Y=Men04 # the second variable (population level) X=Women04 # population size N=length(pik) # joint inclusion probabilities for Poisson sampling pikl=outer(pik,pik,"*") # draw a sample using Poisson sampling s=UPpoisson(pik) # sample inclusion probabilities 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. (sample level) pikls=pikl[s==1,s==1] # ratio estimator and its estimated variance vartaylor_ratio(Ys,Xs,pikls)
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