The Gram–Charlier A series (named in honor of
Jørgen Pedersen Gram
Jørgen Pedersen Gram (27 June 1850 – 29 April 1916) was a Danish actuary and mathematician who was born in Nustrup, Duchy of Schleswig, Denmark and died in Copenhagen, Denmark.
Important papers of his include ''On series expansions determin ...
and
Carl Charlier
Carl Vilhelm Ludwig Charlier (1 April 1862 – 4 November 1934) was a Swedish astronomer. His parents were Emmerich Emanuel and Aurora Kristina (née Hollstein) Charlier.
Career
Charlier was born in Östersund. He received his Ph.D. from ...
), and the Edgeworth series (named in honor of
Francis Ysidro Edgeworth
Francis Ysidro Edgeworth (8 February 1845 – 13 February 1926) was an Anglo-Irish philosopher and political economist who made significant contributions to the methods of statistics during the 1880s. From 1891 onward, he was appointed the ...
) are
series
Series may refer to:
People with the name
* Caroline Series (born 1951), English mathematician, daughter of George Series
* George Series (1920–1995), English physicist
Arts, entertainment, and media
Music
* Series, the ordered sets used in ...
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
cumulant
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will ha ...
s. 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 In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
* The indicator function of a subset, that is the function
::\mathbf_A\colon X \to \,
:which for a given subset ''A'' of ''X'', has value 1 at points ...
of the distribution whose
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
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
cumulant
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will ha ...
s. 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
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
, but other choices are possible as well. By the definition of the cumulants, we have (see Wallace, 1958)
: