VE.Jk.CBS.HT.RegCo.Hajek: The Campbell-Berger-Skinner unequal probability jackknife variance estimator for the estimator of the regression coefficient using the Hajek point estimator (Horvitz-Thompson form)

Description

Computes the Campbell(1980); Berger-Skinner(2005) unequal probability jackknife variance estimator for the estimator of the regression coefficient using the Hajek (1971) point estimator. It uses the Horvitz-Thompson (1952) variance form.

Usage

VE.Jk.CBS.HT.RegCo.Hajek(VecY.s, VecX.s, VecPk.s, MatPkl.s)

Value

The function returns a value for the estimated variance.

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 that 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 that 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.
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

From Linear Regression Analysis, for an imposed population model $$y=\alpha + \beta x$$ the population regression coefficient $\beta$, assuming that the population size $N$ is unknown (see Sarndal et al., 1992, Sec. 5.10), can be estimated by: $$\hat{\beta}_{Hajek} = \frac{\sum_{k\in s} w_k (y_k - \hat{\bar{y}}_{Hajek})(x_k - \hat{\bar{x}}_{Hajek})}{\sum_{k\in s} w_k (x_k - \hat{\bar{x}}_{Hajek})^2}$$ 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 $s$. The variance of $\hat{\beta}_{Hajek}$ can be estimated by the Campbell(1980); Berger-Skinner(2005) unequal probability jackknife variance estimator (implemented by the current function): $$\hat{V}(\hat{\beta}_{Hajek}) = \sum_{k\in s}\sum_{l\in s} \frac{\pi_{kl}-\pi_k\pi_l}{\pi_{kl}} \varepsilon_k \varepsilon_l$$ where $$\varepsilon_k = \left(1-\tilde{w}_k\right) \left(\hat{\beta}_{Hajek}-\hat{\beta}_{Hajek(k)}\right)$$ with $$\tilde{w}_k = \frac{w_k}{\sum_{l\in s} w_l}$$ and where $\hat{\beta}_{Hajek(k)}$ has the same functional form as $\hat{\beta}_{Hajek}$ but omitting the $k$-th element from the sample $s$.

References

Campbell, C. (1980) A different view of finite population estimation. Proceedings of the Survey Research Methods Section of the American Statistical Association, 319--324.

Berger, Y. G. and Skinner, C. J. (2005) A jackknife variance estimator for unequal probability sampling. Journal of the Royal Statistical Society B, 67, 79--89.

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.

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

Examples

Run this code

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
#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 regression coeff. point estimator using y1
VE.Jk.CBS.HT.RegCo.Hajek(y1[s==1], x[s==1], pik.U[s==1], pikl.s)
#Computes the var. est. of the regression coeff. point estimator using y2
VE.Jk.CBS.HT.RegCo.Hajek(y2[s==1], x[s==1], pik.U[s==1], pikl.s)

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