samplingVarEst (version 1.4)

VE.Jk.Tukey.Ratio: The Tukey (1958) jackknife variance estimator for the estimator of a ratio

Description

Computes the Quenouille(1956); Tukey (1958) jackknife variance estimator for the estimator of a ratio of two totals/means.

Usage

VE.Jk.Tukey.Ratio(VecY.s, VecX.s, VecPk.s, N, FPC= TRUE)

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.

N

the population size. It must be an integer or a double-precision scalar with zero-valued fractional part. This information is also utilised for the finite population correction, see FPC below.

FPC

logical value. If an ad hoc finite population correction \(FPC=1-n/N\) is to be used. The default is TRUE.

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 Quenouille(1956); Tukey (1958) jackknife variance estimator (implemented by the current function): $$\hat{V}(\hat{R}) = \left(1-\frac{n}{N}\right)\frac{n-1}{n}\sum_{k\in s} \left( \hat{R}_{(k)}-\hat{R} \right)^2$$ where $$\hat{R}_{(k)} = \frac{\sum_{l\in s, l\neq k} w_l y_l}{\sum_{l\in s, l\neq k} w_l x_l}$$ Note that we are implementing the Tukey (1958) jackknife variance estimator using the `ad hoc' finite population correction \(1-n/N\) (see Shao and Tu, 1995; Wolter, 2007). If FPC=FALSE then the term \(1-n/N\) is ommited from the above formula.

References

Quenouille, M. H. (1956) Notes on bias in estimation. Biometrika, 43, 353--360.

Shao, J. and Tu, D. (1995) The Jackknife and Bootstrap. Springer-Verlag, Inc.

Tukey, J. W. (1958) Bias and confidence in not-quite large samples (abstract). The Annals of Mathematical Statistics, 29, 2, p. 614.

Wolter, K. M. (2007) Introduction to Variance Estimation. 2nd Ed. Springer, Inc.

See Also

VE.Lin.HT.Ratio
VE.Lin.SYG.Ratio
VE.Jk.CBS.HT.Ratio
VE.Jk.CBS.SYG.Ratio
VE.Jk.B.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
N     <- dim(oaxaca)[1]                     #Defines the population size
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.Tukey.Ratio(y1[s==1], x[s==1], pik.U[s==1], N)
#Computes the var. est. of the ratio point estimator using y2
VE.Jk.Tukey.Ratio(y2[s==1], x[s==1], pik.U[s==1], N, FPC= FALSE)