Edgeworth series
In probability theory, 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 terms of its cumulants.
[2] The key idea of these expansions is to write the characteristic function of the distribution whose probability density function f is to be approximated in terms of the characteristic function of a distribution with known and suitable properties, and to recover f through the inverse Fourier transform.
We examine a continuous random variable.
We expand in terms of a known distribution with probability density function ψ, characteristic function
The density ψ 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)[3] which gives the following formal identity: By the properties of the Fourier transform,
, where D is the differential operator with respect to x.
on both sides of the equation, we find for f the formal expansion If ψ is chosen as the normal density with mean and variance as given by f, that is, mean
for all r > 2, as higher cumulants of the normal distribution are 0.
By expanding the exponential and collecting terms according to the order of the derivatives, we arrive at the Gram–Charlier A series.
Such an expansion can be written compactly in terms of Bell polynomials as Since the n-th derivative of the Gaussian function
is given in terms of Hermite polynomial as this gives us the final expression of the Gram–Charlier A series as Integrating the series gives us the cumulative distribution function where
If we include only the first two correction terms to the normal distribution, we obtain with
Note that this expression is not guaranteed to be positive, and is therefore not a valid probability distribution.
The Gram–Charlier A series diverges in many cases of interest—it converges only if
When it does not converge, the series is also not a true asymptotic expansion, because it is not possible to estimate the error of the expansion.
Edgeworth developed a similar expansion as an improvement to the central limit theorem.
[4] The advantage of the Edgeworth series is that the error is controlled, so that it is a true asymptotic expansion.
be a sequence of independent and identically distributed random variables with finite mean
denote the cumulative distribution functions of the variables
, If we expand the formal expression of the characteristic function
in terms of the standard normal distribution, that is, if we set then the cumulant differences in the expansion are The Gram–Charlier A series for the density function of
is now The Edgeworth series is developed similarly to the Gram–Charlier A series, only that now terms are collected according to powers of
The coefficients of n−m/2 term can be obtained by collecting the monomials of the Bell polynomials corresponding to the integer partitions of m. Thus, we have the characteristic function as where
Again, after inverse Fourier transform, the density function
follows as Likewise, integrating the series, we obtain the distribution function We can explicitly write the polynomial
As such we need to examine three cases: Thus, the required polynomial is The first five terms of the expansion are[5] Here, φ(j)(x) is the j-th derivative of φ(·) at point x. Remembering that the derivatives of the density of the normal distribution are related to the normal density by
is the Hermite polynomial of order n), this explains the alternative representations in terms of the density function.
Blinnikov and Moessner (1998) have given a simple algorithm to calculate higher-order terms of the expansion.
Note that in case of a lattice distributions (which have discrete values), the Edgeworth expansion must be adjusted to account for the discontinuous jumps between lattice points.
