Random inequalities and Edgeworth approximation

November 7, 2012
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(This article was originally published at The Endeavour » Statistics, and syndicated at StatsBlogs.)

I’ve written a lot about random inequalities. That’s because computers spend a lot of time computing random inequalities in the inner loop of simulations. I’m looking for ways to speed this up.

Here’s my latest idea: Approximating random inequalities with Edgeworth expansions

An Edgeworth expansion is like a Fourier series, except you use derivatives of the normal density as your basis rather than sine functions. Sometimes the full Edgeworth expansion does not converge and yet the first few terms make a good approximation. The tech report explicitly considers Edgeworth approximations with just two terms, but demonstrates the integration tricks necessary to use more terms. The result is computed in closed form, no numerical integration required, and so may be much faster than other approaches.

One advantage of the Edgeworth approach is that it only depends on the moments of the distributions in the inequality. This means it provides an approximation that’s waiting to be used on new families of distributions. But because it’s not specific to a distribution family, its performance in a particular case needs to be explored. In the case of beta distributions, for example, even a single-term approximation does pretty well.

More blog posts on random inequalities:

Introduction
Analytical results
Numerical results
Cauchy distributions
Beta distributions
Gamma distributions
Three or more random variables
Folded normals
A Bayesian view of Amazon Resellers
Fast approximation of beta inequalities
Shifting probability distributions



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