# Edgeworth Expansion of Discrete Random Variable

Background

Consider a random variable $Z$ given by

$$Z=\sum_{i=1}^nX_i,$$

where the $X_i$ are iid random variables which are linear combinations (with both positive and negative coefficients) of Poisson random variables. In practice $n\approx100$.

I am able to compute the cumulants $\{\kappa_n\}_{n=1}^\infty$ of the variables $X_i$.

I want to (computationally) approximate the value of $F_Z(0)$, where $F_Z$ is the CDF of the random variable $Z$. It seems like the tool of choice is the Edgeworth Expansion.

Questions

1. Is there a reference anywhere for the first 4 or 5 terms of the general Edgeworth expansion of the CDF? On Wikipedia, for example, they give it for the PDF, but I'm stuck trying to integrate that right now.
2. What do I need to do to account for the fact that $Z$ is technically discrete? Some things I have read mention that there need to be adjustments for random variables on the lattice, but I'm not sure how it works for discrete but non-integer-valued random variables.
3. Is this a bad way to approximate $F_Z(0)$? Are there simpler ways? Perhaps a popular library or something?

Disclaimer

Forgive me--I have very little statistics background.

• I'd probably just simulate, say, 10000 instances of $Z$, and use that sample to estimate $F_Z(0)$, myself. It is an interesting question, however. If you are an R user, you can use a package that implements both the PDF and CDF versions of Edgeworth, Gram-Charlier, and Cornish-Fisher expansions: cran.r-project.org/web/packages/PDQutils/README.html. – jbowman Jun 11 '18 at 19:42
• do you really mean the $X_i$ to be iid, or just independent? – jld Jun 11 '18 at 20:23
• @jbowman I will have to investigate those libraries. In general would you expect Monte Carlo to be more accurate than say a 5-term Edgeworth expansion? – David M. Jun 11 '18 at 20:30
• @Chaconne I am happy to assume identically distributed for now, though any more general results would be helpful – David M. Jun 11 '18 at 20:31
• I would expect Monte Carlo to be more accurate, but it would be very interesting to perform the comparison to see how big your sample size has to be to get more accurate. Maybe if I have some free time... other than time spent answering questions on CrossValidated, that is... – jbowman Jun 11 '18 at 20:44