Edgeworth series

The Gram–Charlier A series (named in honor of Jørgen Pedersen Gram

Jørgen Pedersen Gram

Jørgen Pedersen Gram was a Danish actuary and mathematician who was born in Nustrup, Duchy of Schleswig, Denmark and died in Copenhagen, Denmark....

and Carl Charlier

Carl Charlier

Carl Vilhelm Ludwig Charlier was a Swedish astronomer.He received his Ph.D. from Uppsala University in 1887, later worked there and at the Stockholm Observatory and was Professor of Astronomy and Director of the Observatory at Lund University from 1897.He made extensive statistical studies of the...

), and the Edgeworth series (named in honor of Francis Ysidro Edgeworth

Francis Ysidro Edgeworth

Francis Ysidro Edgeworth FBA was an Irish philosopher and political economist who made significant contributions to the methods of statistics during the 1880s...

) are series

Series (mathematics)

A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....

that approximate a probability distribution

Probability distribution

In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

in terms of its cumulant

Cumulant

In probability theory and statistics, the cumulants κn of a probability distribution are a set of quantities that provide an alternative to the moments of the distribution. The moments determine the cumulants in the sense that any two probability distributions whose moments are identical will have...

s. The series are the same; but, the arrangement of terms (and thus the accuracy of truncating the series) differ.

Gram–Charlier A series

The key idea of these expansions is to write the characteristic function

Characteristic function

In mathematics, characteristic function can refer to any of several distinct concepts:* The most common and universal usage is as a synonym for indicator function, that is the function* In probability theory, the characteristic function of any probability distribution on the real line is given by...

of the distribution whose probability density function

Probability density function

In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...

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

Fourier transform

In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

.

Let f be the characteristic function of the distribution whose density function is F, and κ_r its cumulant

Cumulant

s. We expand in terms of a known distribution with probability density function

, characteristic function

, and cumulants γ_r. 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 the following formal identity:

By the properties of the Fourier transform, (it)^rψ(t) is the Fourier transform of (−1)^r D^r

(x), where D is the differential operator with respect to x. Thus, we find for F the formal expansion

is chosen as the normal density with mean and variance as given by F, that is, mean μ = κ₁ and variance σ² = κ₂, then the expansion becomes

By expanding the exponential and collecting terms according to the order of the derivatives, we arrive at the Gram–Charlier A series. If we include only the first two correction terms to the normal distribution, we obtain

with H₃(x) = x³ − 3x and
H₄(x) = x⁴ − 6x² + 3 (these are Hermite polynomials

Hermite polynomials

In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence that arise in probability, such as the Edgeworth series; in combinatorics, as an example of an Appell sequence, obeying the umbral calculus; in numerical analysis as Gaussian quadrature; and in physics, where...

).

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 F(x) falls off faster than exp(−x²/4) at infinity (Cramér 1957). When it does not converge, the series is also not a true asymptotic expansion

Asymptotic expansion

In mathematics an asymptotic expansion, asymptotic series or Poincaré expansion is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular,...

, 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 Gram–Charlier A series.

Edgeworth developed a similar expansion as an improvement to the central limit theorem

Central limit theorem

In probability theory, the central limit theorem states conditions under which the mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed. The central limit theorem has a number of variants. In its common...

. The advantage of the Edgeworth series is that the error is controlled, so that it is a true asymptotic expansion

Asymptotic expansion

.

Let {X_i} be a sequence of independent and identically distributed random variables with mean μ and variance σ², and let Y_n be their standardized sums:

Let F_n denote the cumulative distribution function

Cumulative distribution function

In probability theory and statistics, the cumulative distribution function , or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far"...

s of the variables Y_n. Then by the central limit theorem,

for every x, as long as the mean and variance are finite.

Now assume that the random variables X_i have mean μ, variance σ², and higher cumulants κ_r=σ^rλ_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 Gram–Charlier A series, only that now terms are collected according to powers of n. Thus, we have

where P_j(x) is a polynomial

Polynomial

In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

of degree 3j. Again, after inverse Fourier transform, the density function F_n follows as

The first five terms of the expansion are

Here, Φ^(j)(x) is the j-th derivative of Φ(·) at point x. Blinnikov and Moessner (1998) have given a simple algorithm to calculate higher-order terms of the expansion.

Gram–Charlier A series

Edgeworth series

Further reading