EdgeworthExpansion

An Edgeworth expansion is a technique used in probability theory and statistics to approximate the distribution of a random variable by a series of terms that involve its moments. It was developed by Irish mathematician and economist Francis Ysidro Edgeworth in the late 19th and early 20th centuries. The expansion is particularly useful when the exact distribution of a random variable is unknown or difficult to compute, but its moments (such as mean, variance, skewness, and kurtosis) are known or can be estimated.

The Edgeworth expansion is based on the idea that, given a random variable's moments, one can approximate its probability density function (PDF) or cumulative distribution function (CDF) by using a series involving increasing powers of the standardized variable (i.e., the variable after subtracting its mean and dividing by its standard deviation). The simplest form of the Edgeworth expansion is the Cornish-Fisher expansion, which is primarily used for approximating quantiles.

The Edgeworth expansion has been applied to a wide range of statistical problems, including hypothesis testing, estimation, and confidence intervals. However, it should be noted that the expansion can sometimes provide poor approximations, particularly in the tails of the distribution or when higher moments are not well-behaved. In such cases, alternative techniques like bootstrapping or Monte Carlo methods may be more suitable for approximating the distribution.