VE.Jk.B.Mean.Hajek {samplingVarEst}

The Berger (2007) unequal probability jackknife variance estimator for the Hajek estimator of a mean

Description

Computes the Berger (2007) unequal probability jackknife variance estimator for the Hajek (1971) estimator of a mean.

Usage

VE.Jk.B.Mean.Hajek(VecY.s, 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.

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 Berger (2007) unequal probability jackknife variance estimator (implemented by the current function):

\hat{V}(\hat{\bar{y}}_{Hajek}) = \sum_{k\in s} \frac{n}{n-1}(1-\pi_k) \left(\varepsilon_k - \hat{B}\right)^{2}

where

\hat{B} = \frac{\sum_{k\in s}(1-\pi_k) \varepsilon_k}{\sum_{k\in s}(1-\pi_k)}

and

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

with

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

and

\hat{\bar{y}}_{Hajek(k)} = \frac{\sum_{l\in s, l\neq k} w_l y_l}{\sum_{l\in s, l\neq k} w_l}

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.

Berger, Y. G. (2007) A jackknife variance estimator for unistage stratified samples with unequal probabilities. Biometrika 94, 953–964.

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.

See Also

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