VE_Hajek_Mean_NHT: The Hajek variance estimator for the Narain-Horvitz-Thompson...

In samplingVarEst: Sampling Variance Estimation

The Hajek variance estimator for the Narain-Horvitz-Thompson point estimator for a mean

Description

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

Usage

VE.Hajek.Mean.NHT(VecY.s, VecPk.s, N)

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.
N	the population size. It must be an integer or a double-precision scalar with zero-valued fractional part.

Details

For the population mean of the variable y:

\bar{y} = \frac{1}{N}∑_{k\in U} y_k

the unbiased Narain (1951); Horvitz-Thompson (1952) estimator of \bar{y} is given by:

\hat{\bar{y}}_{NHT} = \frac{1}{N}∑_{k\in s} \frac{y_k}{π_k}

where π_k denotes the inclusion probability of the k-th element in the sample s. For large-entropy sampling designs, the variance of \hat{\bar{y}}_{NHT} is approximated by the Hajek (1964) variance:

V(\hat{\bar{y}}_{NHT}) = \frac{1}{N(N-1)}≤ft[∑_{k\in U}\frac{y_k^2}{π_k}(1-π_k)-dG^2\right]

with d=∑_{k\in U}π_k(1-π_k) and G=d^{-1}∑_{k\in U}(1-π_k)y_k.

The variance V(\hat{t}_{NHT}) can be estimated by the variance estimator (implemented by the current function):

\hat{V}(\hat{\bar{y}}_{NHT}) = \frac{n}{N^2(n-1)}≤ft[∑_{k\in s}≤ft(\frac{y_k}{π_k}\right)^2(1-π_k)-\hat{d}\hat{G}^2\right]

where \hat{d}=∑_{k\in s}(1-π_k) and \hat{G}=\hat{d}^{-1}∑_{k\in s}(1-π)y_k/π_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).

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.

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.Mean.NHT
VE.SYG.Mean.NHT

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
N     <- dim(oaxaca)[1]                     #Defines the population size
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. for y1
VE.Hajek.Mean.NHT(y1[s==1], pik.U[s==1], N)
#Computes the (approximate) var. est. of the NHT point est. for y2
VE.Hajek.Mean.NHT(y2[s==1], pik.U[s==1], N)

samplingVarEst documentation built on Jan. 14, 2023, 5:08 p.m.