samplingVarEst (version 1.4)

Est.RegCoI.Hajek: Estimator of the intercept regression coefficient using the Hajek point estimator

Description

Estimates the population intercept regression coefficient using the Hajek (1971) point estimator.

Usage

Est.RegCoI.Hajek(VecY.s, VecX.s, VecPk.s)

Value

The function returns a value for the intercept regression coefficient point estimator.

Arguments

VecY.s

vector of the variable of interest Y; 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 variable of interest X; 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.

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

From Linear Regression Analysis, for an imposed population model $$y=\alpha + \beta x$$ the population intercept regression coefficient \(\alpha\), assuming that the population size \(N\) is unknown (see Sarndal et al., 1992, Sec. 5.10), can be estimated by: $$\hat{\alpha}_{Hajek} = \hat{\bar{y}}_{Hajek} - \frac{\sum_{k\in s} w_k (y_k - \hat{\bar{y}}_{Hajek})(x_k - \hat{\bar{x}}_{Hajek})}{\sum_{k\in s} w_k (x_k - \hat{\bar{x}}_{Hajek})^2} \hat{\bar{x}}_{Hajek}$$ where \(\hat{\bar{y}}_{Hajek}\) and \(\hat{\bar{x}}_{Hajek}\) are the Hajek (1971) point estimators of the population means \(\bar{y} = N^{-1} \sum_{k\in U} y_k\) and \(\bar{x} = N^{-1} \sum_{k\in U} x_k\), respectively, $$\hat{\bar{y}}_{Hajek} = \frac{\sum_{k\in s} w_k y_k}{\sum_{k\in s} w_k}$$ $$\hat{\bar{x}}_{Hajek} = \frac{\sum_{k\in s} w_k x_k}{\sum_{k\in s} w_k}$$ and \(w_k=1/\pi_k\) with \(\pi_k\) denoting 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.

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

See Also

Est.RegCo.Hajek
VE.Jk.Tukey.RegCoI.Hajek
VE.Jk.CBS.HT.RegCoI.Hajek
VE.Jk.CBS.SYG.RegCoI.Hajek
VE.Jk.B.RegCoI.Hajek
VE.Jk.EB.SW2.RegCoI.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$POP10                       #Defines the variable of interest y1
y2    <- oaxaca$POPMAL10                    #Defines the variable of interest y2
x     <- oaxaca$HOMES10                     #Defines the variable of interest x
#Computes the intercept regression coefficient estimator for y1 and x
Est.RegCoI.Hajek(y1[s==1], x[s==1], pik.U[s==1])
#Computes the intercept regression coefficient estimator for y2 and x
Est.RegCoI.Hajek(y2[s==1], x[s==1], pik.U[s==1])