VE.SYG.Total.NHT {samplingVarEst}

The Sen-Yates-Grundy variance estimator for the Narain-Horvitz-Thompson point estimator for a total

Description

Computes the Sen (1953); Yates-Grundy(1953) variance estimator for the Narain (1951); Horvitz-Thompson (1952) point estimator for a population total.

Usage

VE.SYG.Total.NHT(VecY.s, VecPk.s, MatPkl.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.

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. Let \pi_{kl} denotes the joint-inclusion probabilities of the k-th and l-th elements in the sample s. The variance of \hat{t}_{NHT} is given by:

V(\hat{t}_{NHT}) = \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, if the utilised sampling design is of fixed sample size, can therefore be estimated by the Sen-Yates-Grundy variance estimator (implemented by the current function):

\hat{V}(\hat{t}_{NHT}) = \frac{-1}{2}\sum_{k\in s}\sum_{l\in s} \frac{\pi_{kl}-\pi_k\pi_l}{\pi_{kl}}\left(\frac{y_k}{\pi_k}-\frac{y_l}{\pi_l}\right)^2

Value

The function returns a value for the estimated variance.

Author(s)

Emilio Lopez Escobar.

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.

Sen, A. R. (1953) On the estimate of the variance in sampling with varying probabilities. Journal of the Indian Society of Agricultural Statistics, 5, 119–127.

Yates, F. and Grundy, P. M. (1953) Selection without replacement from within strata with probability proportional to size. Journal of the Royal Statistical Society B, 15, 253–261.

See Also

VE.HT.Total.NHT
VE.Hajek.Total.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
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 var. est. of the NHT point estimator for y1
VE.SYG.Total.NHT(y1[s==1], pik.U[s==1], pikl.s)
#Computes the var. est. of the NHT point estimator for y2
VE.SYG.Total.NHT(y2[s==1], pik.U[s==1], pikl.s)