| VE.Jk.EB.SW2.RegCoI.Hajek {samplingVarEst} | R Documentation |
The self-weighted two-stage sampling Escobar-Berger (2013) jackknife variance estimator for the estimator of the intercept regression coefficient using the Hajek point estimator
Description
Computes the self-weighted two-stage sampling Escobar-Berger (2013) jackknife variance estimator for the estimator of the intercept regression coefficient using the Hajek (1971) point estimator.
Usage
VE.Jk.EB.SW2.RegCoI.Hajek(VecY.s, VecX.s, VecPk.s, nII, VecPi.s,
VecCluLab.s, VecCluSize.s)Arguments
VecY.s |
vector of the variable of interest Y; its length is equal to |
VecX.s |
vector of the variable of interest X; its length is equal to |
VecPk.s |
vector of the elements' first-order inclusion probabilities; its length is equal to |
nII |
the second stage sample size, i.e. the fixed number of ultimate sampling units that were selected within each cluster. Its size must be less than or equal to the minimum cluster size in the sample. |
VecPi.s |
vector of the clusters' first-order inclusion probabilities; its length is equal to |
VecCluLab.s |
vector of the clusters' labels for the elements; its length is equal to |
VecCluSize.s |
vector of the clusters' sizes; its length is equal to |
Details
From Linear Regression Analysis, for an imposed population model
y=\alpha + \beta x
the population intercept regression coefficient \alpha, assuming that the population size N is unknown (see Sarndal et al., 1992, Sec. 5.10), can be estimated by:
\hat{\alpha}_{Hajek} = \hat{\bar{y}}_{Hajek} - \frac{\sum_{k\in s} w_k (y_k - \hat{\bar{y}}_{Hajek})(x_k - \hat{\bar{x}}_{Hajek})}{\sum_{k\in s} w_k (x_k - \hat{\bar{x}}_{Hajek})^2} \hat{\bar{x}}_{Hajek}
where \hat{\bar{y}}_{Hajek} and \hat{\bar{x}}_{Hajek} are the Hajek (1971) point estimators of the population means \bar{y} = N^{-1} \sum_{k\in U} y_k and \bar{x} = N^{-1} \sum_{k\in U} x_k, respectively,
\hat{\bar{y}}_{Hajek} = \frac{\sum_{k\in s} w_k y_k}{\sum_{k\in s} w_k}
\hat{\bar{x}}_{Hajek} = \frac{\sum_{k\in s} w_k x_k}{\sum_{k\in s} w_k}
and w_k=1/\pi_k with \pi_k denoting the inclusion probability of the k-th element in the sample s. If s is a self-weighted two-stage sample, the variance of \hat{\alpha}_{Hajek} can be estimated by the Escobar-Berger (2013) jackknife variance estimator (implemented by the current function):
\hat{V}(\hat{\alpha}_{Hajek}) = v_{clu} + v_{obs}
v_{clu} = \sum_{i\in s} (1-\pi_{Ii}^{*}) \varsigma_{(Ii)}^{2} - \frac{1}{\hat{d}}\left(\sum_{i\in s} (1-\pi_{Ii}) \varsigma_{(Ii)}\right)^{2}
v_{obs} = \sum_{k\in s} \phi_k \varepsilon_{(k)}^{2}
where \hat{d}={\sum}_{i\in s}{(1-\pi_{Ii})}, \phi_k = I\{k\in s_{i}\}\pi_{Ii}^{*}(M_{i}-n_{II})/(M_{i}-1), \pi_{Ii}^{*} = \pi_{Ii}n_{II}(M_{i}-1)/(n_{II}-1)M_{i}, with s_{i} denoting the sample elements from the i-th cluster, I\{k\in s_{i}\} is an indicator that takes the value 1 if the k-th observation is within the i-th cluster and 0 otherwise, \pi_{Ii} is the inclusion probability of the i-th cluster in the sample s, M_{i} is the size of the i-th cluster, n_{II} is the sample size within each cluster, n_{I} is the number of sampled clusters, and where
\varsigma_{(Ii)}=\frac{n_{I}-1}{n_{I}} (\hat{\alpha}_{Hajek}-\hat{\alpha}_{Hajek(Ii)})
\varepsilon_{(k)}=\frac{n-1}{n} (\hat{\alpha}_{Hajek}-\hat{\alpha}_{Hajek(k)})
where \hat{\alpha}_{Hajek(Ii)} and \hat{\alpha}_{Hajek(k)} have the same functional form as \hat{\alpha}_{Hajek} but omitting the i-th cluster and the k-th element, respectively, from the sample s.
Note that this variance estimator implicitly utilises the Hajek (1964) approximations that are designed for large-entropy sampling designs, large samples, and large populations, i.e., care should be taken with highly-stratified samples, e.g. Berger (2005).
Value
The function returns a value for the estimated variance.
Author(s)
Emilio Lopez Escobar.
References
Berger, Y. G. (2005) Variance estimation with highly stratified sampling designs with unequal probabilities. Australian & New Zealand Journal of Statistics, 47, 365–373.
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. (1964) Asymptotic theory of rejective sampling with varying probabilities from a finite population. The Annals of Mathematical Statistics, 35, 4, 1491–1523.
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.
Sarndal, C.-E. and Swensson, B. and Wretman, J. (1992) Model Assisted Survey Sampling. Springer-Verlag, Inc.
See Also
VE.Jk.EB.SW2.RegCo.HajekVE.Jk.Tukey.RegCoI.HajekVE.Jk.CBS.HT.RegCoI.HajekVE.Jk.CBS.SYG.RegCoI.HajekVE.Jk.B.RegCoI.Hajek
Examples
data(oaxaca) #Loads the Oaxaca municipalities dataset
s <- oaxaca$sSW_10_3 #Defines the sample to be used
SampData <- oaxaca[s==1, ] #Defines the sample dataset
nII <- 3 #Defines the 2nd stage fixed sample size
CluLab.s <- SampData$IDDISTRI #Defines the clusters' labels
CluSize.s <- SampData$SIZEDIST #Defines the clusters' sizes
piIi.s <- (10 * CluSize.s / 570) #Reconstructs clusters' 1st order incl. probs.
pik.s <- piIi.s * (nII/CluSize.s) #Reconstructs elements' 1st order incl. probs.
y1.s <- SampData$POP10 #Defines the variable y1
y2.s <- SampData$POPMAL10 #Defines the variable y2
x.s <- SampData$HOMES10 #Defines the variable x
#Computes the var. est. of the intercept reg. coeff. point estimator using y1
VE.Jk.EB.SW2.RegCoI.Hajek(y1.s, x.s, pik.s, nII, piIi.s, CluLab.s, CluSize.s)
#Computes the var. est. of the intercept reg. coeff. point estimator using y2
VE.Jk.EB.SW2.RegCoI.Hajek(y2.s, x.s, pik.s, nII, piIi.s, CluLab.s, CluSize.s)