I am looking for strategies how to approximate poisson-binomial distribution (PB) via the binomial (B) distribution. I have seen a few papers [Ehm91,Roos01,LeCam59] on them.
The papers uses total variation and arithmetic means. I am interested in upper bounding the CDF of PB.
I ran simulations for more than a day comparing $F_{PB}(k,n;\mathbf{p})$ and $F_{B}(k,n;p_g)$ where $p_g$ is the geometric mean (instead of the usual arithmetic mean) of the vector of $\mathbf{p}\in[0,1]^n$ with random. Throughout my simulations, with random $\mathbf{p}$ and various $n\leq 15$ (max acceptable runtime for my computer). It is always $F_{PB}(k)\leq F_{B}(k)$ for all $k\in[0,n]$.
It is simple to show the tails is true as $ F_{PB}(n-1) = F_{B}(n-1;p_g) $.
I have a few questions
- Is $F_{PB}(k)\leq F_{B}(k;p_g)$ true for all $k$ and $n$?
- If so, is there a way to prove it ?
- else, is there a counter-example ?
Thank you