Let's say we have a random variable $Y$ defined as the sum of $N$ Bernoulli variables $X_i$, each with a different, success probability $p_i$ and a different (fixed) weight $w_i$. The weights are positive and between ~0.001-1,000 with a handful of outliers. If it would make the problem more tractable, it would be OK to discard both the very large and very small outliers.
Formally, $Y = \sum X_i w_i$
Where $\Pr(X_i=1)=p_i$ and $\Pr(X_i=0)=1-p_i$.
$N$ can range from ~10-10,000. I would like to quickly compute a approximation to $\Pr(Y<=k)$ (where $k$ is given).
The data I'm handling comes from real life bets, each with implied odds $p_i$ and wager $w_i$. Here are plots showing the rough distribution of the probabilities and weights:
One way to compute the approximation is through Monte Carlo simulation, but that can get slow for large values of $N$. One can also use the Central Limit Theorem to compute this for large $N$, but the accuracy is quite poor for small $N$ (or if there are a handful of large weights). Finally, there are ways to improve on the CLT approach by using asymptoptic expansions, e.g.:
Volkova, A. Y. (1996). A refinement of the central limit theorem for sums of independent random indicators. Theory of Probability and its Applications 40, 791-794.
However, as far as I've seen, the refined approximations only specify how to compute $Y$ if all weights $W_i$ are equal to 1 (a standard Poisson Binomial distribution). Is there a way to compute an answer that works for small and large $N$ with a smaller error than a pure CLT approach without having to resort to Monte Carlo Simulation? In other words, is it possible to extend the improved closed-form approximations to allow weights not equal to 1?
Some related questions that don't quite answer mine:
- CLT can be used for weighted sum of different Bernoulli variables? (Pure CLT is too inaccurate)
- How to find the distribution of the weighted sum of independent Bernoulli random variables for positive non-integer weights (No answer)
- https://math.stackexchange.com/questions/1546366/distribution-of-weighted-sum-of-bernoulli-rvs (The bounds mentioned in [Raghavan, 1988] are very, very wide for small N or when computing the CDF near the mean)