VE.Jk.B.RegCoI.Hajek: The Berger (2007) unequal probability jackknife variance estimator for the estimator of the intercept regression coefficient using the Hajek point estimator

Description

Computes the Berger (2007) unequal probability jackknife variance estimator for the estimator of the intercept regression coefficient using the Hajek (1971) point estimator.

Usage

VE.Jk.B.RegCoI.Hajek(VecY.s, VecX.s, VecPk.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 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 $s$. The variance of $\hat{\alpha}_{Hajek}$ can be estimated by the Berger (2007) unequal probability jackknife variance estimator (implemented by the current function): $$\hat{V}(\hat{\alpha}_{Hajek}) = \sum_{k\in s} \frac{n}{n-1}(1-\pi_k) \left(\varepsilon_k - \hat{B}\right)^{2}$$ where $$\hat{B} = \frac{\sum_{k\in s}(1-\pi_k) \varepsilon_k}{\sum_{k\in s}(1-\pi_k)}$$ and $$\varepsilon_k = \left(1-\tilde{w}_k\right) \left(\hat{\alpha}_{Hajek}-\hat{\alpha}_{Hajek(k)}\right)$$ with $$\tilde{w}_k = \frac{w_k}{\sum_{l\in s} w_l}$$ and where $\hat{\alpha}_{Hajek(k)}$ has the same functional form as $\hat{\alpha}_{Hajek}$ but omitting the $k$-th element from the sample $s$. Note that this variance estimator utilises implicitly the Hajek (1964) approximations that are designed for large-entropy sampling designs, large samples and large populations, i.e. care should be taken with highly-stratified samples, e.g. Berger (2005).

References

Berger, Y. G. (2005) Variance estimation with highly stratified sampling designs with unequal probabilities. Australian & New Zealand Journal of Statistics, 47, 365--373.

Berger, Y. G. (2007) A jackknife variance estimator for unistage stratified samples with unequal probabilities. Biometrika 94, 953--964.

Hajek, J. (1964) Asymptotic theory of rejective sampling with varying probabilities from a finite population. The Annals of Mathematical Statistics, 35, 4, 1491--1523.

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.

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
#Computes the var. est. of the intercept reg. coeff. point estimator using y1
VE.Jk.B.RegCoI.Hajek(y1[s==1], x[s==1], pik.U[s==1])
#Computes the var. est. of the intercept reg. coeff. point estimator using y2
VE.Jk.B.RegCoI.Hajek(y2[s==1], x[s==1], pik.U[s==1])

Run the code above in your browser using DataLab