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)
# }