# VE_SYG_Mean_NHT: The Sen-Yates-Grundy variance estimator for the... In samplingVarEst: Sampling Variance Estimation

 VE.SYG.Mean.NHT R Documentation

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

### Description

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

### Usage

VE.SYG.Mean.NHT(VecY.s, VecPk.s, MatPkl.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. 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. 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. Let π_{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}∑_{k\in U}∑_{l\in U} (π_{kl}-π_kπ_l)\frac{y_k}{π_k}\frac{y_l}{π_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{\bar{y}}_{NHT}) = \frac{1}{N^2}\frac{-1}{2}∑_{k\in s}∑_{l\in s} \frac{π_{kl}-π_kπ_l}{π_{kl}}≤ft(\frac{y_k}{π_k}-\frac{y_l}{π_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.

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


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