samplingVarEst (version 1.5)

VE.Lin.HT.Ratio: The unequal probability linearisation variance estimator for the estimator of a ratio (Horvitz-Thompson form)

Description

Computes the unequal probability Taylor linearisation variance estimator for the estimator of a ratio of two totals/means. It uses the Horvitz-Thompson (1952) variance form.

Usage

VE.Lin.HT.Ratio(VecY.s, VecX.s, VecPk.s, MatPkl.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 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.

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 unequal probability linearisation variance estimator (implemented by the current function). For details see Woodruff (1971); Deville (1999); Demnati-Rao (2004); Sarndal et al., (1992, Secs. 5.5 and 5.6): $$\hat{V}(\hat{R}) = \sum_{k\in s}\sum_{l\in s} \frac{\pi_{kl}-\pi_k\pi_l}{\pi_{kl}} w_k u_k w_l u_l$$ where $$u_k = \frac{y_k - \hat{R} x_k}{\hat{t}_{x,NHT}}$$ with $$\hat{t}_{x,NHT} = \sum_{k\in s} w_k x_k$$ the unbiased Narain (1951); Horvitz-Thompson (1952) estimator of the population total for the (denominator) variable VecX.s.

References

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

Deville, J.-C. (1999) Variance estimation for complex statistics and estimators: linearization and residual techniques. Survey Methodology, 25, 193--203.

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.

Sarndal, C.-E. and Swensson, B. and Wretman, J. (1992) Model Assisted Survey Sampling. Springer-Verlag, Inc.

Woodruff, R. S. (1971) A Simple Method for Approximating the Variance of a Complicated Estimate. Journal of the American Statistical Association, 66, 334, 411--414.

See Also

VE.Lin.SYG.Ratio
VE.Jk.Tukey.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
y1     <- oaxaca$POP10                       #Defines the numerator variable y1
y2     <- oaxaca$POPMAL10                    #Defines the numerator variable y2
x      <- oaxaca$HOMES10                     #Defines the denominator variable x
#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.Lin.HT.Ratio(y1[s==1], x[s==1], pik.U[s==1], pikl.s)
#Computes the var. est. of the ratio point estimator using y2
VE.Lin.HT.Ratio(y2[s==1], x[s==1], pik.U[s==1], pikl.s)