optimStrat (version 2.1)

varpips: Variance of Pareto PIps Sampling with the HT Estimator

Description

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

Usage

varpips(y,x,n)

Arguments

y

a numeric vector giving the values of the study variable.

x

a positive numeric vector giving the values of the auxiliary variable that is used in order to define the inclusion probabilities.

n

a positive integer indicating the desired sample size.

Value

A numeric value giving the variance of the Horvitz-Thompson 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 Horvitz-Thompson 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 $$t_{1} = \sum\frac{y_{k}^{2}(1-\pi_{k})}{\pi_{k}}$$ $$t_{2} = \sum y_{k}(1-\pi_{k})$$ $$t_{3} = \sum \pi_{k}(1-\pi_{k})$$

References

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

See Also

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; varpipsreg for the variance of the regression estimator under probability proportional-to-size 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)
varpips(y,x=x^1.25,n=500)
# }