samplingVarEst (version 1.4)

Est.Ratio: Estimator of a ratio

Description

Estimates a population ratio of two totals/means.

Usage

Est.Ratio(VecY.s, VecX.s, VecPk.s)

Value

The function returns a value for the ratio point estimator.

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 \(R\) (implemented by the current function) 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 \(s\).

References

Hajek, J. (1971) Comment on An essay on the logical foundations of survey sampling by Basu, D. in Foundations of Statistical Inference (Godambe, V.P. and Sprott, D.A. eds.), p. 236. Holt, Rinehart and Winston.

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.

See Also

VE.Jk.Tukey.Ratio
VE.Jk.CBS.HT.Ratio
VE.Jk.CBS.SYG.Ratio
VE.Jk.B.Ratio
VE.Jk.EB.SW2.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
Est.Ratio(y1[s==1], x[s==1], pik.U[s==1])   #Ratio estimator for y1 and x
Est.Ratio(y2[s==1], x[s==1], pik.U[s==1])   #Ratio estimator for y2 and x