samplingVarEst (version 1.4)

VE.Hajek.Total.NHT: The Hajek variance estimator for the Narain-Horvitz-Thompson point estimator for a total

Description

Computes the Hajek (1964) variance estimator for the Narain (1951); Horvitz-Thompson (1952) point estimator for a population total.

Usage

VE.Hajek.Total.NHT(VecY.s, VecPk.s)

Value

The function returns a value for the estimated variance.

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 the length 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.

Author

Emilio Lopez Escobar.

Details

For the population total of the variable \(y\): $$t = \sum_{k\in U} y_k$$ the unbiased Narain (1951); Horvitz-Thompson (1952) estimator of \(t\) is given by: $$\hat{t}_{NHT} = \sum_{k\in s} \frac{y_k}{\pi_k}$$ where \(\pi_k\) denotes the inclusion probability of the \(k\)-th element in the sample \(s\). For large-entropy sampling designs, the variance of \(\hat{t}_{NHT}\) is approximated by the Hajek (1964) variance: $$V(\hat{t}_{NHT}) = \frac{N}{N-1}\left[\sum_{k\in U}\frac{y_k^2}{\pi_k}(1-\pi_k)-dG^2\right]$$ with \(d=\sum_{k\in U}\pi_k(1-\pi_k)\) and \(G=d^{-1}\sum_{k\in U}(1-\pi_k)y_k\).

The variance \(V(\hat{t}_{NHT})\) can be estimated by the variance estimator (implemented by the current function): $$\hat{V}(\hat{t}_{NHT}) = \frac{n}{n-1}\left[\sum_{k\in s}\left(\frac{y_k}{\pi_k}\right)^2(1-\pi_k)-\hat{d}\hat{G}^2\right]$$ where \(\hat{d}=\sum_{k\in s}(1-\pi_k)\) and \(\hat{G}=\hat{d}^{-1}\sum_{k\in s}(1-\pi)y_k/\pi_k\).

Note that the Hajek (1964) variance approximation is 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).

References

Berger, Y. G. (2005) Variance estimation with highly stratified sampling designs with unequal probabilities. Australian & New Zealand Journal of Statistics, 47, 365--373.

Hajek, J. (1964) Asymptotic theory of rejective sampling with varying probabilities from a finite population. The Annals of Mathematical Statistics, 35, 4, 1491--1523.

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.

Narain, R. D. (1951) On sampling without replacement with varying probabilities. Journal of the Indian Society of Agricultural Statistics, 3, 169--175.

See Also

VE.HT.Total.NHT
VE.SYG.Total.NHT

Examples

data(oaxaca)                                 #Loads the Oaxaca municipalities dataset
pik.U <- Pk.PropNorm.U(373, oaxaca$SURFAC05) #Reconstructs the 1st order incl. probs.
s     <- oaxaca$sSURFAC                      #Defines the sample to be used
y1    <- oaxaca$POP10                        #Defines the variable of interest y1
y2    <- oaxaca$HOMES10                      #Defines the variable of interest y2
#Computes the (approximate) var. est. of the NHT point est. from y1
VE.Hajek.Total.NHT(y1[s==1], pik.U[s==1])
#Computes the (approximate) var. est. of the NHT point est. from y2
VE.Hajek.Total.NHT(y2[s==1], pik.U[s==1])