optimStrat (version 2.1)

varpipsreg: Design variance of a PIps--reg sampling strategy.

Description

Compute the design variance of the regression estimator of the total of y under Pareto probability proportional-to-size Sampling, where the size variable is indicated by x_des and the sample size is n.

Usage

varpipsreg(y, x_des, n, x_est)

Arguments

y

a numeric vector giving the values of the study variable.

x_des

a positive numeric vector giving the values of the auxiliary variable that is used for defining the inclusion probabilities.

n

a positive integer indicating the desired sample size.

x_est

a positive numeric vector giving the values of the auxiliary variable that is used at the estimation stage.

Value

A numeric value giving the variance of the regression estimator under Pareto probability proportional-to-size Sampling.

Details

Target inclusion probabilities are computed as \(\pi_{k} = n\cdot x_{k}/\sum x_{k}\).

If \(\pi_{k}>1\) for at least one element, \(\pi_k\) is set equal to one for those elements and the inclusion probabilities are calculated again for the remaining elements with the remaining sample size.

Once the \(\pi_k\) are obtained, the variance of the poststratified estimator under Pareto probability proportional-to-size Sampling is computed as: \(V_{\pi ps}\left[\hat{t}_{HT}\right] = \frac{N}{N-1}(t_{1}-\frac{t_{2}^{2}}{t_{3}})\) with $$t_{1} = \sum\frac{E_{k}^{2}(1-\pi_{k})}{\pi_{k}}$$ $$t_{2} = \sum E_{k}(1-\pi_{k})$$ $$t_{3} = \sum \pi_{k}(1-\pi_{k})$$ with \(E_{k}=y_{k}-\hat{y}_{k}\).

References

Rosen, B. (1997). On Sampling with Probability Proportional to Size. Journal of Statistical Planning and Inference 62, 159-191.

See Also

varpips for the variance of the Horvitz-Thompson estimator under probability proportional-to-size sampling; varstsi for the variance of the Horvitz-Thompson estimator under stratified simple random sampling; varpipspos for the variance of the poststratified estimator under probability proportional-to-size sampling; varstsipos for the variance of the poststratified estimator under stratified simple random sampling; varstsireg for the variance of the regression estimator under stratified simple random sampling.

Examples

# NOT RUN {
x<- 1 + sort( rgamma(5000, shape=4/9, scale=108) ) #simulating the auxiliary variable
y<- simulatey(x,b0=10,b1=1,b2=1.25,b4=0.75,rho=0.95)
varpipsreg(y, x_des=x^0.75, n=500, x_est=x^1.25)
# }