The primary units selected with probabilities proportional to size:
\(p_i=M_i/M\)
The Hansen-Hurwitz (p.p.s.) estimator is:
\(\hat{\tau}_p=\dfrac{M}{n}\sum\limits_{i=1}^n \left(\dfrac{y_i}{M_i}\right)\)
Denote by \(\bar{y}_i=\dfrac{y_i}{M_i}\)
\(\hat{V}ar(\hat{\tau}_p)=\dfrac{M^2}{n(n-1)}\sum\limits_{i=1}^n (\bar{y}_i-\hat{\mu}_p)^2\) where
\(\hat{\mu}_p=\dfrac{\hat{\tau}_p}{M}\) is unbiased for \(\mu\).
Thus we also see that:
\(\hat{V}ar(\hat{\mu}_p)=\dfrac{1}{n(n-1)}\sum\limits_{i=1}^n (\bar{y}_i-\hat{\mu}_p)^2\)
Example: Estimating population mean per secondary unit when primary units are selected by pps
From the "Total number of computer help requests" example in Lesson 3.1, 3 clusters out of 10 clusters are sampled (n = 3) with replacement. The data are:
\(y_1=420, y_2 = 1785, y_3=2198\)
\(M_1=650, M_2=2840, M_3=3200\)
Try it!
\begin{align}
\hat{\mu}_p &= \dfrac{1}{n} \sum\limits_{i=1}^n \dfrac{y_i}{M_i}\\
&= \dfrac{1}{3}\times \left(\dfrac{420}{650}+\dfrac{1785}{2840}+\dfrac{2198}{3200}\right)\\
&= 0.6538\\
\end{align}
\begin{align}
\hat{V}ar(\hat{\mu}_p)&=\dfrac{1}{n(n-1)}\sum\limits_{i=1}^n (\bar{y}_i-\hat{\mu}_p)^2\\
&= \dfrac{1}{3 \times 2}[(0.6462-0.6538)^2+(0.6285-0.6538)^2+(0.6869-0.6538)^2]\\
&= 0.000299\\
\end{align}