samplingVarEst (version 1.4)

Est.Mean.Hajek: The Hajek estimator for a mean

Description

Computes the Hajek (1971) estimator for a population mean.

Usage

Est.Mean.Hajek(VecY.s, VecPk.s)

Value

The function returns a value for the mean point estimator.

Arguments

VecY.s

vector of the 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. There must not be missing values.

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 mean of the variable \(y\): $$\bar{y} = \frac{1}{N} \sum_{k\in U} y_k$$ the approximately unbiased Hajek (1971) estimator of \(\bar{y}\) (implemented by the current function) is given by: $$\hat{\bar{y}}_{Hajek} = \frac{\sum_{k\in s} w_k y_k}{\sum_{k\in s} w_k}$$ 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.

See Also

Est.Mean.NHT
VE.Jk.Tukey.Mean.Hajek
VE.Jk.CBS.HT.Mean.Hajek
VE.Jk.CBS.SYG.Mean.Hajek
VE.Jk.B.Mean.Hajek
VE.Jk.EB.SW2.Mean.Hajek

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$POP10                       #Defines the variable of interest y1
y2    <- oaxaca$HOMES10                     #Defines the variable of interest y2
Est.Mean.Hajek(y1[s==1], pik.U[s==1])       #Computes the Hajek est. for y1
Est.Mean.Hajek(y2[s==1], pik.U[s==1])       #Computes the Hajek est. for y2