VE.EB.HT.Mean.Hajek {samplingVarEst}

The Escobar-Berger unequal probability replicate variance estimator for the Hajek (1971) estimator of a mean (Horvitz-Thompson form)

Description

Computes the Escobar-Berger (2013) unequal probability replicate variance estimator for the Hajek estimator of a mean. It uses the Horvitz-Thompson (1952) variance form.

Usage

VE.EB.HT.Mean.Hajek(VecY.s, VecPk.s, MatPkl.s,
                    VecAlpha.s = rep.int(1, length(VecPk.s)))

Arguments

VecY.s

vector of the variable of interest; its length is equal to n, the 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 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.

VecAlpha.s

vector of the \alpha_k values; its length is equal to n, the sample size. Values in VecAlpha.s can be different for each unit, and must be greater or equal to zero. Escobar-Berger (2013) showed that this replicate variance estimator is valid for \alpha_k\geq 0. In particular, they suggest using \alpha_k=1 for all units in the sample (the default for VecAlpha.s if omitted in the function call). Using \alpha_k>1 approximates the Demnati-Rao (2004) linearisation variance estimators. There must not be missing values.

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 Escobar-Berger (2013) unequal probability replicate variance estimator (implemented by the current function):

\hat{V}(\hat{\bar{y}}_{Hajek}) = \sum_{k\in s}\sum_{l\in s} \frac{\pi_{kl}-\pi_k\pi_l}{\pi_{kl}} \breve{\nu}_k \breve{\nu}_l

where

\breve{\nu}_k = w_k^{\alpha_k} \left(\hat{\bar{y}}_{Hajek}-\hat{\bar{y}}_{Hajek,k}^{*}\right)

for some \alpha_k\geq0 (suggested to be 1, see below comments) and with

\hat{\bar{y}}_{Hajek,k}^{*} = \frac{\sum_{l\in s} w_l y_l - w_k^{1-\alpha_k} y_k}{\sum_{l\in s} w_l - w_k^{1-\alpha_k}}

Regarding the value of \alpha_k, Escobar-Berger (2013) show that \hat{V}(\hat{\bar{y}}_{Hajek}) is valid for \alpha_k\geq0 but conclude that \alpha_k>0 should be used as \alpha_k=0 corresponds to a naive biased and unstable jackknife. They recommend \alpha_k=1 or \alpha_k>1. If \alpha_k=1, \hat{V}(\hat{\bar{y}}_{Hajek}) reduces to the Escobar-Berger (2011) jackknife. Using \alpha_k>1 approximates the empirical influence function, i.e. the Gateaux (1919) derivative, or Demnati-Rao (2004) linearisation variance estimators. The larger the \alpha_k, the closer the approximation. Further, Escobar-Berger (2013) give an intuitive explanation of the replication method from a jackknife and bootstrap perspective.

Value

The function returns a value for the estimated variance.

Author(s)

Emilio Lopez Escobar.

References

Demnati, A. and Rao, J. N. K. (2004) Linearization variance estimators for survey data. Survey Methodology, 30, 17–26.

Escobar, E. L. and Berger, Y. G. (2011) Jackknife variance estimation for functions of Horvitz-Thompson estimators under unequal probability sampling without replacement. In Proceeding of the 58th World Statistics Congress. Dublin, Ireland: International Statistical Institute.

Escobar, E. L. and Berger, Y. G. (2013) A new replicate variance estimator for unequal probability sampling without replacement. Canadian Journal of Statistics 41, 3, 508–524.

Gateaux, R. (1919) Fonctions d'une infinite de variables indeependantes. Bulletin de la Societe Mathematique de France, 47, 70–96.

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.

See Also

VE.Jk.Tukey.Mean.Hajek
VE.Jk.CBS.SYG.Mean.Hajek
VE.Jk.B.Mean.Hajek
VE.Jk.EB.SW2.Mean.Hajek
VE.EB.SYG.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$POP10                       #Defines the variable y1
y2      <- oaxaca$POPMAL10                    #Defines the variable y2
Alpha.s <- rep(2, times=373)                  #Defines the vector with Alpha values
#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.EB.HT.Mean.Hajek(y1[s==1], pik.U[s==1], pikl.s)
#Computes the var. est. of the Hajek mean point estimator using y2
VE.EB.HT.Mean.Hajek(y2[s==1], pik.U[s==1], pikl.s, Alpha.s)