samplingVarEst (version 1.4)

VE.Jk.B.Ratio: The Berger (2007) unequal probability jackknife variance estimator for the estimator of a ratio

Description

Computes the Berger (2007) unequal probability jackknife variance estimator for the estimator of a ratio of two totals/means.

Usage

VE.Jk.B.Ratio(VecY.s, VecX.s, VecPk.s)

Value

The function returns a value for the estimated variance.

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 the length 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 the length 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.

Author

Emilio Lopez Escobar.

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 Berger (2007) unequal probability jackknife variance estimator (implemented by the current function): $$\hat{V}(\hat{R}) = \sum_{k\in s} \frac{n}{n-1}(1-\pi_k) \left(\varepsilon_k - \hat{B}\right)^{2}$$ where $$\hat{B} = \frac{\sum_{k\in s}(1-\pi_k) \varepsilon_k}{\sum_{k\in s}(1-\pi_k)}$$ and $$\varepsilon_k = \left(1-\tilde{w}_k\right) \left(\hat{R}-\hat{R}_{(k)}\right)$$ with $$\tilde{w}_k = \frac{w_k}{\sum_{l\in s} w_l}$$ and $$\hat{R}_{(k)} = \frac{\sum_{l\in s, l\neq k} w_l y_l/\sum_{l\in s, l\neq k} w_l}{\sum_{l\in s, l\neq k} w_l x_l/\sum_{l\in s, l\neq k} w_l} = \frac{\sum_{l\in s, l\neq k} w_l y_l}{\sum_{l\in s, l\neq k} w_l x_l}$$ Note that this variance estimator utilises implicitly the Hajek (1964) approximations that are 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).

References

Berger, Y. G. (2005) Variance estimation with highly stratified sampling designs with unequal probabilities. Australian & New Zealand Journal of Statistics, 47, 365--373.

Berger, Y. G. (2007) A jackknife variance estimator for unistage stratified samples with unequal probabilities. Biometrika 94, 953--964.

Hajek, J. (1964) Asymptotic theory of rejective sampling with varying probabilities from a finite population. The Annals of Mathematical Statistics, 35, 4, 1491--1523.

See Also

VE.Lin.HT.Ratio
VE.Lin.SYG.Ratio
VE.Jk.Tukey.Ratio
VE.Jk.CBS.HT.Ratio
VE.Jk.CBS.SYG.Ratio
VE.Jk.EB.SW2.Ratio
VE.EB.HT.Ratio
VE.EB.SYG.Ratio

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
#Computes the var. est. of the ratio point estimator using y1
VE.Jk.B.Ratio(y1[s==1], x[s==1], pik.U[s==1])
#Computes the var. est. of the ratio point estimator using y2
VE.Jk.B.Ratio(y2[s==1], x[s==1], pik.U[s==1])