R: The self-weighted two-stage sampling Escobar-Berger (2013)...

VE.Jk.EB.SW2.Total.Hajek {samplingVarEst}

R Documentation

The self-weighted two-stage sampling Escobar-Berger (2013) jackknife variance estimator for the Hajek (1971) estimator of a total

Description

Computes the self-weighted two-stage sampling Escobar-Berger (2013) jackknife variance estimator for the Hajek estimator of a total.

Usage

VE.Jk.EB.SW2.Total.Hajek(VecY.s, VecPk.s, nII, VecPi.s,
                         VecCluLab.s, VecCluSize.s, N)

Arguments

`VecY.s`	vector of the variable of interest; its length is equal to `n`, the total sample size. Its length has to be the same as that of `VecPk.s`. There must not be missing values.
`VecPk.s`	vector of the elements' first-order inclusion probabilities; its length is equal to `n`, the total sample size. Values in `VecPk.s` must be greater than zero and less than or equal to one. There must not be missing values.
`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 `n`, the total sample size. Hence values are expected to be repeated in the utilised sample dataset. Values in `VecPi.s` must be greater than zero and less than or equal to one. There must not be missing values.
`VecCluLab.s`	vector of the clusters' labels for the elements; its length is equal to `n`, the total sample size. The labels must be integer numbers.
`VecCluSize.s`	vector of the clusters' sizes; its length is equal to `n`, the total sample size. Hence values are expected to be repeated in the utilised sample dataset. None of the sizes must be smaller than `nII`.
`N`	the population size. It must be an integer or a double-precision scalar with zero-valued fractional part.

Details

For the population total of the variable y:

t = \sum_{k\in U} y_k

the approximately unbiased Hajek (1971) estimator of t is given by:

\hat{t}_{Hajek} = N \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. If s is a self-weighted two-stage sample, the variance of \hat{t}_{Hajek} can be estimated by the Escobar-Berger (2013) jackknife variance estimator (implemented by the current function):

\hat{V}(\hat{t}_{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{t}_{Hajek}-\hat{t}_{Hajek(Ii)})

\varepsilon_{(k)}=\frac{n-1}{n} (\hat{t}_{Hajek}-\hat{t}_{Hajek(k)})

where \hat{t}_{Hajek(Ii)} and \hat{t}_{Hajek(k)} have the same functional form as \hat{t}_{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.

Examples

data(oaxaca)                          #Loads the Oaxaca municipalities dataset
s         <- oaxaca$sSW_10_3          #Defines the sample to be used
N         <- dim(oaxaca)[1]           #Defines the population size
SampData  <- oaxaca[s==1, ]           #Defines the sample dataset
nII       <- 3                        #Defines the 2nd stage fixed sample size
CluLab.s  <- SampData$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 of interest y1
y2.s      <- SampData$POPMAL10        #Defines the variable of interest y2
#Computes the var. est. of the Hajek total point estimator using y1
VE.Jk.EB.SW2.Total.Hajek(y1.s, pik.s, nII, piIi.s, CluLab.s, CluSize.s, N)
#Computes the var. est. of the Hajek total point estimator using y2
VE.Jk.EB.SW2.Total.Hajek(y2.s, pik.s, nII, piIi.s, CluLab.s, CluSize.s, N)

[Package samplingVarEst version 1.5 Index]