R: The Escobar-Berger unequal probability replicate variance...

VE.EB.SYG.Ratio {samplingVarEst}

R Documentation

The Escobar-Berger unequal probability replicate variance estimator for the estimator of a ratio (Sen-Yates-Grundy form)

Description

Computes the Escobar-Berger (2013) unequal probability replicate variance estimator for the estimator of a ratio of two totals/means. It uses the Sen (1953); Yates-Grundy(1953) variance form.

Usage

VE.EB.SYG.Ratio(VecY.s, VecX.s, VecPk.s, MatPkl.s,
                VecAlpha.s = rep.int(1, length(VecPk.s)))

Arguments

`VecY.s`	vector of the numerator variable of interest; its length is equal to `n`, the sample size. Its length has to be the same as that of `VecPk.s` and `VecX.s`. There must not be missing values.
`VecX.s`	vector of the denominator variable of interest; its length is equal to `n`, the sample size. Its length has to be the same as that of `VecPk.s` and `VecY.s`. There must not be missing values. All values of `VecX.s` should be greater than zero. A warning is displayed if this does not hold, and computations continue if mathematical expressions allow this kind of values for the denominator variable.
`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.
`VecAlpha.s`	vector of the `\alpha_k` values; its length is equal to `n`, the sample size. Values in `VecAlpha.s` can be different for each unit, and must be greater or equal to zero. Escobar-Berger (2013) showed that this replicate variance estimator is valid for `\alpha_k\geq 0`. In particular, they suggest using `\alpha_k=1` for all units in the sample (the default for `VecAlpha.s` if omitted in the function call). Using `\alpha_k>1` approximates the Demnati-Rao (2004) linearisation variance estimators. There must not be missing values.

Details

For the population ratio of two totals/means of the variables y and x:

R = \frac{\sum_{k\in U} y_k/N}{\sum_{k\in U} x_k/N} = \frac{\sum_{k\in U} y_k}{\sum_{k\in U} x_k}

the ratio estimator of R is given by:

\hat{R} = \frac{\sum_{k\in s} w_k y_k}{\sum_{k\in s} w_k x_k}

where w_k=1/\pi_k and \pi_k denotes the inclusion probability of the k-th element in the sample s. The variance of \hat{R} can be estimated by the Escobar-Berger (2013) unequal probability replicate variance estimator (implemented by the current function):

\hat{V}(\hat{R}) = \frac{-1}{2}\sum_{k\in s}\sum_{l\in s} \frac{\pi_{kl}-\pi_k\pi_l}{\pi_{kl}} (\breve{\nu}_k - \breve{\nu}_l)^{2}

where

\breve{\nu}_k = w_k^{\alpha_k} \left(\hat{R}-\hat{R}_k^{*}\right)

for some \alpha_k\geq0 (suggested to be 1, see below comments) and with

\hat{R}_k^{*} = \frac{\left(\sum_{l\in s} w_l y_l - w_k^{1-\alpha_k} y_k\right)/\left(\sum_{l\in s} w_l - w_k^{1-\alpha_k} \right)}{\left(\sum_{l\in s} w_l x_l - w_k^{1-\alpha_k} x_k\right)/\left(\sum_{l\in s} w_l - w_k^{1-\alpha_k} \right)} = \frac{\sum_{l\in s} w_l y_l - w_k^{1-\alpha_k} y_k}{\sum_{l\in s} w_l x_l - w_k^{1-\alpha_k} x_k}

Regarding the value of \alpha_k, Escobar-Berger (2013) show that \hat{V}(\hat{R}) is valid for \alpha_k\geq0 but conclude that \alpha_k>0 should be used as \alpha_k=0 corresponds to a naive biased and unstable jackknife. They recommend \alpha_k=1 or \alpha_k>1. If \alpha_k=1, \hat{V}(\hat{R}) reduces to the Escobar-Berger (2011) jackknife. Using \alpha_k>1 approximates the empirical influence function, i.e. the Gateaux (1919) derivative, or Demnati-Rao (2004) linearisation variance estimators. The larger the \alpha_k, the closer the approximation. Further, Escobar-Berger (2013) give an intuitive explanation of the replication method from a jackknife and bootstrap perspective.

Value

The function returns a value for the estimated variance.

Author(s)

Emilio Lopez Escobar.

References

Demnati, A. and Rao, J. N. K. (2004) Linearization variance estimators for survey data. Survey Methodology, 30, 17–26.

Escobar, E. L. and Berger, Y. G. (2011) Jackknife variance estimation for functions of Horvitz-Thompson estimators under unequal probability sampling without replacement. In Proceeding of the 58th World Statistics Congress. Dublin, Ireland: International Statistical Institute.

Escobar, E. L. and Berger, Y. G. (2013) A new replicate variance estimator for unequal probability sampling without replacement. Canadian Journal of Statistics 41, 3, 508–524.

Gateaux, R. (1919) Fonctions d'une infinite de variables indeependantes. Bulletin de la Societe Mathematique de France, 47, 70–96.

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.

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$POP10                       #Defines the numerator variable y1
y2      <- oaxaca$POPMAL10                    #Defines the numerator variable y2
x       <- oaxaca$HOMES10                     #Defines the denominator variable x
Alpha.s <- rep(2, times=373)                  #Defines the vector with Alpha values
#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 ratio point estimator using y1
VE.EB.SYG.Ratio(y1[s==1], x[s==1], pik.U[s==1], pikl.s) #Using default VecAlpha.s
#Computes the var. est. of the ratio point estimator using y2
VE.EB.SYG.Ratio(y2[s==1], x[s==1], pik.U[s==1], pikl.s, Alpha.s)

[Package samplingVarEst version 1.5 Index]