| VE.Jk.CBS.SYG.Mean.Hajek {samplingVarEst} | R Documentation |
The Campbell-Berger-Skinner unequal probability jackknife variance estimator for the Hajek (1971) estimator of a mean (Sen-Yates-Grundy form)
Description
Computes the Campbell(1980); Berger-Skinner(2005) unequal probability jackknife variance estimator for the Hajek estimator of a mean. It uses the Sen (1953); Yates-Grundy(1953) variance form.
Usage
VE.Jk.CBS.SYG.Mean.Hajek(VecY.s, VecPk.s, MatPkl.s)Arguments
VecY.s |
vector of the variable of interest; its length is equal to |
VecPk.s |
vector of the first-order inclusion probabilities; its length is equal to |
MatPkl.s |
matrix of the second-order inclusion probabilities; its number of rows and columns equals |
Details
For the population mean of the variable y:
\bar{y} = \frac{1}{N} \sum_{k\in U} y_k
the approximately unbiased Hajek (1971) estimator of \bar{y} is given by:
\hat{\bar{y}}_{Hajek} = \frac{\sum_{k\in s} w_k y_k}{\sum_{k\in s} w_k}
where w_k=1/\pi_k and \pi_k denotes the inclusion probability of the k-th element in the sample s. The variance of \hat{\bar{y}}_{Hajek} can be estimated by the Campbell(1980); Berger-Skinner(2005) unequal probability jackknife variance estimator (implemented by the current function):
\hat{V}(\hat{\bar{y}}_{Hajek}) = \frac{-1}{2}\sum_{k\in s}\sum_{l\in s} \frac{\pi_{kl}-\pi_k\pi_l}{\pi_{kl}} (\varepsilon_k - \varepsilon_l)^{2}
where
\varepsilon_k = \left(1-\tilde{w}_k\right) \left(\hat{\bar{y}}_{Hajek}-\hat{\bar{y}}_{Hajek(k)}\right)
with
\tilde{w}_k = \frac{w_k}{\sum_{l\in s} w_l}
and
\hat{\bar{y}}_{Hajek(k)} = \frac{\sum_{l\in s, l\neq k} w_l y_l}{\sum_{l\in s, l\neq k} w_l}
The Sen-Yates-Grundy form for the Campbell(1980); Berger-Skinner(2005) unequal probability jackknife variance estimator is proposed in Escobar-Berger (2013) under less-restrictive regularity conditions.
Value
The function returns a value for the estimated variance.
Author(s)
Emilio Lopez Escobar.
References
Campbell, C. (1980) A different view of finite population estimation. Proceedings of the Survey Research Methods Section of the American Statistical Association, 319–324.
Berger, Y. G. and Skinner, C. J. (2005) A jackknife variance estimator for unequal probability sampling. Journal of the Royal Statistical Society B, 67, 79–89.
Escobar, E. L. and Berger, Y. G. (2013) A jackknife variance estimator for self-weighted two-stage samples. Statistica Sinica, 23, 595–613.
Hajek, J. (1971) Comment on An essay on the logical foundations of survey sampling by Basu, D. in Foundations of Statistical Inference (Godambe, V.P. and Sprott, D.A. eds.), p. 236. Holt, Rinehart and Winston.
Sen, A. R. (1953) On the estimate of the variance in sampling with varying probabilities. Journal of the Indian Society of Agricultural Statistics, 5, 119–127.
Yates, F. and Grundy, P. M. (1953) Selection without replacement from within strata with probability proportional to size. Journal of the Royal Statistical Society B, 15, 253–261.
See Also
VE.Jk.Tukey.Mean.HajekVE.Jk.CBS.HT.Mean.HajekVE.Jk.B.Mean.HajekVE.Jk.EB.SW2.Mean.Hajek
Examples
data(oaxaca) #Loads the Oaxaca municipalities dataset
pik.U <- Pk.PropNorm.U(373, oaxaca$HOMES00) #Reconstructs the 1st order incl. probs.
s <- oaxaca$sHOMES00 #Defines the sample to be used
y1 <- oaxaca$POP10 #Defines the variable of interest y1
y2 <- oaxaca$POPMAL10 #Defines the variable of interest y2
#This approximation is only suitable for large-entropy sampling designs
pikl.s <- Pkl.Hajek.s(pik.U[s==1]) #Approx. 2nd order incl. probs. from s
#Computes the var. est. of the Hajek mean point estimator using y1
VE.Jk.CBS.SYG.Mean.Hajek(y1[s==1], pik.U[s==1], pikl.s)
#Computes the var. est. of the Hajek mean point estimator using y2
VE.Jk.CBS.SYG.Mean.Hajek(y2[s==1], pik.U[s==1], pikl.s)