The GramCharlier A series and the Edgeworth series, named in honor of Francis Ysidro Edgeworth, are series that approximate a probability distribution in terms of its cumulants. The series are the same; but, the arrangement of terms (and thus the accuracy of truncating the series) differ.
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The key idea of these expansions is to write the characteristic function of the distribution whose probability density function is F 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.
Let f be the characteristic function of the distribution whose density function is F, and κ_{r} its cumulants. We expand in terms of a known distribution with probability density function Ψ, characteristic function ψ, and standardized cumulants γ_{r}. 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 the following formal identity:
By the properties of the Fourier transform, (it)^{r}ψ(t) is the Fourier transform of (−1)^{r} D^{r} Ψ(x), where D is the differential operator with respect to x. Thus, 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 μ = κ_{1} and variance σ^{2} = κ_{2}, then the expansion becomes
By expanding the exponential and collecting terms according to the order of the derivatives, we arrive at the GramCharlier A series. If we include only the first two correction terms to the normal distribution, we obtain
with H_{3}(x) = x^{3} − 3x and H_{4}(x) = x^{4} − 6x^{2} + 3 (these are Hermite polynomials).
Note that this expression is not guaranteed to be positive, and is therefore not a valid probability distribution. The GramCharlier A series diverges in many cases of interest—it converges only if F(x) falls off faster than exp(−x^{2}/4) at infinity (Cramér 1957). 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. For this reason, the Edgeworth series (see next section) is generally preferred over the GramCharlier A series.
Edgeworth developed a similar expansion as an improvement to the central limit theorem. The advantage of the Edgeworth series is that the error is controlled, so that it is a true asymptotic expansion.
Let {X_{i}} be a sequence of independent and identically distributed random variables with means μ and variances σ^{2}, and let Y_{n} be their standardized sums:
Denote F_{n} the cumulative distribution functions of the variables Y_{n}. Then by the central limit theorem,
for every x, as long as the means and variances are finite and the sum of variances diverges to infinity.
Now assume that the random variables X_{i} have mean μ, variance σ^{2}, and higher cumulants κ_{r}=σ^{r}λ_{r}. If we expand in terms of the unit normal distribution, that is, if we set
then the cumulant differences in the formal expression of the characteristic function f_{n}(t) of F_{n} are
The Edgeworth series is developed similarly to the GramCharlier A series, only that now terms are collected according to powers of n. Thus, we have
where P_{j}(x) is a polynomial of degree 3j. Again, after inverse Fourier transform, the density function F_{n} follows as
The first five terms of the expansion are ^{[1]}
Here, Φ^{(j)}(x) is the jth derivative of Φ(·) at point x. Blinnikov and Moessner (1998) have given a simple algorithm to calculate higherorder terms of the expansion.
