Edgeworth series

Edgeworth series

The Gram-Charlier 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.

Gram-Charlier A series

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 Psi, characteristic function psi, and standardized cumulants γ"r". The density Psi 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:

:f(t)=expleft [sum_{r=1}^infty(kappa_r-gamma_r)frac{(it)^r}{r!} ight] psi(t),.

By the properties of the Fourier transform, ("it")"r"ψ("t") is the Fourier transform of (−1)"r" "D""r" Psi("x"), where "D" is the differential operator with respect to "x". Thus, we find for "F" the formal expansion

:F(x) = expleft [sum_{r=1}^infty(kappa_r - gamma_r)frac{(-D)^r}{r!} ight] Psi(x),.

If Psi 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

:F(x) = expleft [sum_{r=3}^inftykappa_rfrac{(-D)^r}{r!} ight] frac{1}{sqrt{2pi}sigma}expleft [-frac{(x-mu)^2}{2sigma^2} ight] ,.

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

: F(x) = frac{1}{sqrt{2pi}sigma}expleft [-frac{(x-mu)^2}{2sigma^2} ight] left [1+frac{kappa_3}{3!sigma^3}H_3left(frac{x-mu}{sigma} ight)+frac{kappa_4}{4!sigma^4}H_4left(frac{x-mu}{sigma} ight) ight] ,,

with "H"3("x") = "x"3 − 3"x" and "H"4("x") = "x"4 − 6"x"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 Gram-Charlier 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 Gram-Charlier A series.

Edgeworth 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 identically distributed random variables, and "Y""n" the standardized sum

: Y_n = frac{sum_{i=1}^n(X_i-mbox{E} [X_i] )}{sqrt{sum_{i=1}^nmbox{var} [X_i] .

Further, let "F""n" be the probability density function of the variables "Y""n". By the central limit theorem,

:lim_{n ightarrowinfty} F_n(x) = frac{1}{sqrt{2piexp(- frac{1}{2}x^2)

for every "x", as long as the means and variances are finite and the sum of variances diverges to infinity. (Generally, the conclusion of the central limit theorem is about the limit of cumulative distribution functions, not of probability density functions, and therefore applies to discrete distributions as well. But discrete distributions are not contemplated in the present context).

Now assume that the random variables "X""i" have mean μ, variance σ2, and higher cumulants κ"r"=σ"r"λ"r". If we expand in terms of the unit normal distribution, that is, if we set

:Psi(x)=frac{1}{sqrt{2piexp(- frac{1}{2}x^2)

then the cumulant differences in the formal expression of the characteristic function "f""n"(t) of "F""n" are

: kappa_1-gamma_1 = 0,,

: kappa_2-gamma_2 = 0,,

: kappa_r-gamma_r = frac{lambda_r}{n^{r/2-1; qquad rgeq 3,.

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

: f_n(t)=left [1+sum_{j=1}^infty frac{P_j(it)}{n^{j/2 ight] exp(-t^2/2),,

where "P""j"("x") is a polynomial of degree 3"j". Again, after inverse Fourier transform, the density function "F""n" follows as

: F_n(x) = Psi(x) + sum_{j=1}^infty frac{P_j(-D)}{n^{j/2 Psi(x),.

The first three terms of the expansion are (Cramér 1957)

: F_n(x)approx Psi(x) - frac{lambda_3 Psi^{(3)}(x)}{6sqrt{n +frac{1}{n}left [frac{lambda_4 Psi^{(4)}(x)}{24}+frac{lambda_3^2 Psi^{(6)}(x)}{72} ight] +O(1/n^{3/2}),.

Here, Psi("j")("x") is the "j"th derivative of Psi("x") with respect to "x". Blinnikov and Moessner (1998) have given a simple algorithm to calculate higher-order terms of the expansion.

Further reading

* H. Cramér (1957). "Mathematical Methods of Statistics". Princeton University Press, Princeton.
* D. L. Wallace (1958). "Asymptotic approximations to distributions". "Ann. Math. Stat." 29:635-654.
* P. McCullagh (1987). "Tensor Methods in Statistics". Chapman and Hall, London.
* D. R. Cox and O. E. Barndorff-Nielsen (1989). "Asymptotic Techniques for Use in Statistics". Chapman and Hall, London.
* P. Hall (1992). "The Bootstrap and Edgeworth Expansion". Springer, New York.
* S. Blinnikov and R. Moessner (1998). "Expansions for nearly Gaussian distributions". "Astron. Astrophys. Suppl. Ser." 130:193-205.
* J. E. Kolassa (2006). "Series Approximation Methods in Statistics, 3rd Edition". (Lecture Notes in Statistics #88). Springer, New York.

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