I'm currently trying to learn how to calculate the Horwitz - Thompson estimator for population variances. Using this formula
$$ \hat{V}ar(\hat{\tau}_\pi)=\sum\limits_{i=1}^v \left( \dfrac{1-\pi_i}{\pi^2_i} \right) y^2_i + \sum\limits_{i=1}^v \sum\limits_{j\neq i} \left( \dfrac{\pi_{ij}-\pi_i \pi_j}{\pi_i \pi_j}\right)\dfrac{1}{\pi_{ij}} y_i y_j $$
from the Penn State STAT-506 page, I tried setting up the equation and inputting the provided probabilities and values. However, I was unable to calculate the correct answer. I did notice though if I multiplied the second section $$..+ \sum\limits_{i=1}^v \sum\limits_{j\neq i} \left( \dfrac{\pi_{ij}-\pi_i \pi_j}{\pi_i \pi_j}\right)\dfrac{1}{\pi_{ij}} y_i y_j $$ of the equation by 2 in this way
$$ \hat{V}ar(\hat{\tau}_\pi)=\sum\limits_{i=1}^v \left( \dfrac{1-\pi_i}{\pi^2_i} \right) y^2_i + 2[\sum\limits_{i=1}^v \sum\limits_{j\neq i} \left( \dfrac{\pi_{ij}-\pi_i \pi_j}{\pi_i \pi_j}\right)\dfrac{1}{\pi_{ij}} y_i y_j] $$
I was able to calculate the correct answer.
I'm new to reading these kinds of statistical formulas so I'm wondering is there a term in the initial equation that is telling me it needs to be multiplied by 2? Or am I misreading the equation entirely.
Thanks for all any help
As per Whubers question, here is an example of my calculations:
$\pi_i$= Probability of inclusion of ith sampling unit. $p_1$ = 0.01, $p_2$ = 0.05, $p_3$ = 0.02
$\pi_{ij}$ = Probability of inlclusion of ith sampling unit given the jth sampling unit has been chosen.
$\pi_{12}$ = 0.00565, $\pi_{13}$ = 0.00229, $\pi_{23}$ = 0.01115
$y_i$ = size of the ith sampling unit. $y_1$ = 14 , $y_2$ = 50 , $y_3$ = 25
So for my first set of pairwise calculations ( samples 1 & 2), my calculation was as follows:
$\frac{0.00565 - 0.0394(0.1855)}{0.0394(0.1855)}(\frac{1}{0.00565})(14)(50)$
