samplingVarEst (version 1.4)

VE.HT.Mean.NHT: The Horvitz-Thompson variance estimator for the Narain-Horvitz-Thompson point estimator for a mean

Description

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

Usage

VE.HT.Mean.NHT(VecY.s, VecPk.s, MatPkl.s, N)

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.

MatPkl.s

matrix of the second-order inclusion probabilities; its number of rows and columns is equal to \(n\), 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.

N

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

Author

Emilio Lopez Escobar.

Details

For the population mean of the variable \(y\): $$\bar{y} = \frac{1}{N}\sum_{k\in U} y_k$$ the unbiased Narain (1951); Horvitz-Thompson (1952) estimator of \(\bar{y}\) is given by: $$\hat{\bar{y}}_{NHT} = \frac{1}{N}\sum_{k\in s} \frac{y_k}{\pi_k}$$ where \(\pi_k\) denotes the inclusion probability of the \(k\)-th element in the sample \(s\). Let \(\pi_{kl}\) denotes the joint-inclusion probabilities of the \(k\)-th and \(l\)-th elements in the sample \(s\). The variance of \(\hat{\bar{y}}_{NHT}\) is given by: $$V(\hat{\bar{y}}_{NHT}) = \frac{1}{N^2}\sum_{k\in U}\sum_{l\in U} (\pi_{kl}-\pi_k\pi_l)\frac{y_k}{\pi_k}\frac{y_l}{\pi_l}$$ which can therefore be estimated by the Horvitz-Thompson variance estimator (implemented by the current function): $$\hat{V}(\hat{\bar{y}}_{NHT}) = \frac{1}{N^2}\sum_{k\in s}\sum_{l\in s} \frac{\pi_{kl}-\pi_k\pi_l}{\pi_{kl}}\frac{y_k}{\pi_k}\frac{y_l}{\pi_l}$$

References

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.SYG.Mean.NHT
VE.Hajek.Mean.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
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
#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 variance estimation of the NHT point estimator for y1
VE.HT.Mean.NHT(y1[s==1], pik.U[s==1], pikl.s, N)
#Computes the variance estimation of the NHT point estimator for y2
VE.HT.Mean.NHT(y2[s==1], pik.U[s==1], pikl.s, N)