VE.Jk.EB.SW2.Corr.Hajek {samplingVarEst}R Documentation

The self-weighted two-stage sampling Escobar-Berger (2013) jackknife variance estimator for the estimator of a correlation coefficient using the Hajek point estimator

Description

Computes the self-weighted two-stage sampling Escobar-Berger (2013) jackknife variance estimator for the estimator of a correlation coefficient of two variables using the Hajek (1971) point estimator.

Usage

VE.Jk.EB.SW2.Corr.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 nn, the total sample size. Its length has to be the same as that of VecPk.s and VecX.s. There must not be missing values.

VecX.s

vector of the variable of interest X; its length is equal to nn, the total sample size. Its length has to be the same as that of VecPk.s and VecY.s. There must not be missing values.

VecPk.s

vector of the elements' first-order inclusion probabilities; its length is equal to nn, 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 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 nn, 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 nn, the total sample size. The labels must be integer numbers.

VecCluSize.s

vector of the clusters' sizes; its length is equal to nn, 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.

Details

For the population correlation coefficient of two variables yy and xx:

C=kU(ykyˉ)(xkxˉ)kU(ykyˉ)2kU(xkxˉ)2C = \frac{\sum_{k\in U} (y_k - \bar{y})(x_k - \bar{x})}{\sqrt{\sum_{k\in U} (y_k - \bar{y})^2}\sqrt{\sum_{k\in U} (x_k - \bar{x})^2}}

the point estimator of CC, assuming that NN is unknown (see Sarndal et al., 1992, Sec. 5.9), is:

C^Hajek=kswk(ykyˉ^Hajek)(xkxˉ^Hajek)kswk(ykyˉ^Hajek)2kswk(xkxˉ^Hajek)2\hat{C}_{Hajek} = \frac{\sum_{k\in s} w_k (y_k - \hat{\bar{y}}_{Hajek})(x_k - \hat{\bar{x}}_{Hajek})}{\sqrt{\sum_{k\in s} w_k (y_k - \hat{\bar{y}}_{Hajek})^2}\sqrt{\sum_{k\in s} w_k (x_k - \hat{\bar{x}}_{Hajek})^2}}

where yˉ^Hajek\hat{\bar{y}}_{Hajek} is the Hajek (1971) point estimator of the population mean yˉ=N1kUyk\bar{y} = N^{-1} \sum_{k\in U} y_k,

yˉ^Hajek=kswkykkswk\hat{\bar{y}}_{Hajek} = \frac{\sum_{k\in s} w_k y_k}{\sum_{k\in s} w_k}

and wk=1/πkw_k=1/\pi_k with πk\pi_k denoting the inclusion probability of the kk-th element in the sample ss. If ss is a self-weighted two-stage sample, the variance of C^Hajek\hat{C}_{Hajek} can be estimated by the Escobar-Berger (2013) jackknife variance estimator (implemented by the current function):

V^(C^Hajek)=vclu+vobs\hat{V}(\hat{C}_{Hajek}) = v_{clu} + v_{obs}

vclu=is(1πIi)ς(Ii)21d^(is(1πIi)ς(Ii))2v_{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}

vobs=ksϕkε(k)2v_{obs} = \sum_{k\in s} \phi_k \varepsilon_{(k)}^{2}

where d^=is(1πIi)\hat{d}={\sum}_{i\in s}{(1-\pi_{Ii})}, ϕk=I{ksi}πIi(MinII)/(Mi1)\phi_k = I\{k\in s_{i}\}\pi_{Ii}^{*}(M_{i}-n_{II})/(M_{i}-1), πIi=πIinII(Mi1)/(nII1)Mi\pi_{Ii}^{*} = \pi_{Ii}n_{II}(M_{i}-1)/(n_{II}-1)M_{i}, with sis_{i} denoting the sample elements from the ii-th cluster, I{ksi}I\{k\in s_{i}\} is an indicator that takes the value 11 if the kk-th observation is within the ii-th cluster and 00 otherwise, πIi\pi_{Ii} is the inclusion probability of the ii-th cluster in the sample ss, MiM_{i} is the size of the ii-th cluster, nIIn_{II} is the sample size within each cluster, nIn_{I} is the number of sampled clusters, and where

ς(Ii)=nI1nI(C^HajekC^Hajek(Ii))\varsigma_{(Ii)}=\frac{n_{I}-1}{n_{I}} (\hat{C}_{Hajek}-\hat{C}_{Hajek(Ii)})

ε(k)=n1n(C^HajekC^Hajek(k))\varepsilon_{(k)}=\frac{n-1}{n} (\hat{C}_{Hajek}-\hat{C}_{Hajek(k)})

where C^Hajek(Ii)\hat{C}_{Hajek(Ii)} and C^Hajek(k)\hat{C}_{Hajek(k)} have the same functional form as C^Hajek\hat{C}_{Hajek} but omitting the ii-th cluster and the kk-th element, respectively, from the sample ss. 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.Tukey.Corr.Hajek
VE.Jk.CBS.HT.Corr.Hajek
VE.Jk.CBS.SYG.Corr.Hajek
VE.Jk.B.Corr.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 corr. coeff. point estimator using y1
VE.Jk.EB.SW2.Corr.Hajek(y1.s, x.s, pik.s, nII, piIi.s, CluLab.s, CluSize.s)
#Computes the var. est. of the corr. coeff. point estimator using y2
VE.Jk.EB.SW2.Corr.Hajek(y2.s, x.s, pik.s, nII, piIi.s, CluLab.s, CluSize.s)

[Package samplingVarEst version 1.5 Index]