VE.Jk.CBS.HT.Corr.Hajek {samplingVarEst}

The Campbell-Berger-Skinner unequal probability jackknife variance estimator for the estimator of a correlation coefficient using the Hajek point estimator (Horvitz-Thompson form)

Description

Computes the Campbell(1980); Berger-Skinner(2005) unequal probability jackknife variance estimator for the estimator of a correlation coefficient of two variables using the Hajek (1971) point estimator. It uses the Horvitz-Thompson (1952) variance form.

Usage

VE.Jk.CBS.HT.Corr.Hajek(VecY.s, VecX.s, VecPk.s, MatPkl.s)

Arguments

VecY.s

vector of the variable of interest Y; its length is equal to n, the 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 n, the 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 first-order inclusion probabilities; its length is equal to n, the sample size. Values in VecPk.s must be greater than zero and less than or equal to one. There must not be missing values.

MatPkl.s

matrix of the second-order inclusion probabilities; its number of rows and columns equals n, the sample size. Values in MatPkl.s must be greater than zero and less than or equal to one. There must not be missing values.

Details

For the population correlation coefficient of two variables y and x:

C = \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 C, assuming that N is unknown (see Sarndal et al., 1992, Sec. 5.9), is:

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

\hat{\bar{y}}_{Hajek} = \frac{\sum_{k\in s} w_k y_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. The variance of \hat{C}_{Hajek} can be estimated by the Campbell(1980); Berger-Skinner(2005) unequal probability jackknife variance estimator (implemented by the current function):

\hat{V}(\hat{C}_{Hajek}) = \sum_{k\in s}\sum_{l\in s} \frac{\pi_{kl}-\pi_k\pi_l}{\pi_{kl}} \varepsilon_k \varepsilon_l

where

\varepsilon_k = \left(1-\tilde{w}_k\right) \left(\hat{C}_{Hajek}-\hat{C}_{Hajek(k)}\right)

with

\tilde{w}_k = \frac{w_k}{\sum_{l\in s} w_l}

and where \hat{C}_{Hajek(k)} has the same functional form as \hat{C}_{Hajek} but omitting the k-th element from the sample s.

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.

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.

Horvitz, D. G. and Thompson, D. J. (1952) A generalization of sampling without replacement from a finite universe. Journal of the American Statistical Association, 47, 663–685.

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.SYG.Corr.Hajek
VE.Jk.B.Corr.Hajek
VE.Jk.EB.SW2.Corr.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
x      <- oaxaca$HOMES10                     #Defines the variable of interest x
#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 corr. coeff. point estimator using y1
VE.Jk.CBS.HT.Corr.Hajek(y1[s==1], x[s==1], pik.U[s==1], pikl.s)
#Computes the var. est. of the corr. coeff. point estimator using y2
VE.Jk.CBS.HT.Corr.Hajek(y2[s==1], x[s==1], pik.U[s==1], pikl.s)