The Gram–Charlier A series (named in honor of
Jørgen Pedersen Gram and
Carl Charlier), and the Edgeworth series (named in honor of
Francis Ysidro Edgeworth) are
series that approximate a
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
in terms of its
cumulants. The series are the same; but, the arrangement of terms (and thus the accuracy of truncating the series) differ. The key idea of these expansions is to write the
characteristic function of the distribution whose
probability density function is to be approximated in terms of the characteristic function of a distribution with known and suitable properties, and to recover through the inverse
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
.
Gram–Charlier A series
We examine a continuous random variable. Let
be the characteristic function of its distribution whose density function is , and
its
cumulants. We expand in terms of a known distribution with probability density function , characteristic function
, and cumulants
. The density is generally chosen to be that of the
normal distribution, but other choices are possible as well. By the definition of the cumulants, we have (see Wallace, 1958)
: