I would need to estimate the Bernoulli proportion using non-probability sampling framework. Namely, assume an experiment that selects from an array $A$, $n$ successive success proportions $u_i = r_i / n_i$, i.e., $A = \{(r_1,n_1), (r_2,n_2)...(r_m,n_m)\}$, where $n << m$. Each proportion is assigned a probability $p_i$, so that the array $A$ is ordered in descending order w.r.t. $p_i$. That is, the $n$ elements that are chosen are not randomly selected but from the ordered array $A$, i.e., $p_1 > p_2 > p_3 >...>p_m$. Notice, that it does not mean that $u_i > u_j$ for some $i < j$.

Am I allowed to use the standard Wilson method to estimate the proportion using the $n$ selected elements, i.e,

$\hat{p} = \frac{0.5*\kappa^2 + \sum_{i=1}^{n}{r_i}}{\kappa^2 + \sum_{i=1}^{n}{n_i}}$, where $\kappa = \Phi^{-1}(1-\alpha/2)$,

or should I use the Horvitz-Thompson correction for each $r_i$, i.e., $\tilde{r}_i = r_i/p_i$, where $r_i$ is the number of successes in the $i$th position of the array $A$?


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