Est.RegCoI.Hajek | R Documentation |
Estimates the population intercept regression coefficient using the Hajek (1971) point estimator.
Est.RegCoI.Hajek(VecY.s, VecX.s, VecPk.s)
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 that of |
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 that of |
VecPk.s |
vector of the first-order inclusion probabilities; its length is equal to n, the sample size. Values in |
From Linear Regression Analysis, for an imposed population model
y=α + β x
the population intercept regression coefficient α, assuming that the population size N is unknown (see Sarndal et al., 1992, Sec. 5.10), can be estimated by:
\hat{α}_{Hajek} = \hat{\bar{y}}_{Hajek} - \frac{∑_{k\in s} w_k (y_k - \hat{\bar{y}}_{Hajek})(x_k - \hat{\bar{x}}_{Hajek})}{∑_{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} ∑_{k\in U} y_k and \bar{x} = N^{-1} ∑_{k\in U} x_k, respectively,
\hat{\bar{y}}_{Hajek} = \frac{∑_{k\in s} w_k y_k}{∑_{k\in s} w_k}
\hat{\bar{x}}_{Hajek} = \frac{∑_{k\in s} w_k x_k}{∑_{k\in s} w_k}
and w_k=1/π_k with π_k denoting the inclusion probability of the k-th element in the sample s.
The function returns a value for the intercept regression coefficient point estimator.
Emilio Lopez Escobar.
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.
Est.RegCo.Hajek
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VE.Jk.CBS.HT.RegCoI.Hajek
VE.Jk.CBS.SYG.RegCoI.Hajek
VE.Jk.B.RegCoI.Hajek
VE.Jk.EB.SW2.RegCoI.Hajek
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])
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