World Library  
Flag as Inappropriate
Email this Article

Edgeworth series

Article Id: WHEBN0000641132
Reproduction Date:

Title: Edgeworth series  
Author: World Heritage Encyclopedia
Language: English
Subject: Cumulant, Statistical theory, Mathematical series, Lattice model (finance), Berry–Esseen theorem
Collection: Mathematical Series, Statistical Approximations, Statistical Theory
Publisher: World Heritage Encyclopedia

Edgeworth series

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.[1] The series are the same; but, the arrangement of terms (and thus the accuracy of truncating the series) differ.[2]


  • Gram–Charlier A series 1
  • The Edgeworth series 2
  • Illustration: density of the sample mean of 3 Χ² 3
  • Disadvantages of the Edgeworth expansion 4
  • See also 5
  • References 6
  • Further reading 7

Gram–Charlier A series

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. Let f be the characteristic function of its distribution whose density function is F, and \kappa_r its cumulants. We expand in terms of a known distribution with probability density function Ψ, characteristic function ψ, and cumulants \gamma_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 (see Wallace, 1958)[3]

f(t)= \exp\left[\sum_{r=1}^\infty\kappa_r\frac{(it)^r}{r!}\right] and

which gives the following formal identity:


By the properties of the Fourier transform, (it)^r\psi(t) is the Fourier transform of (-1)^r[D^r\Psi](-x), where D is the differential operator with respect to x. Thus, after changing x with -x on both sides of the equation, we find for F the formal expansion

F(x) = \exp\left[\sum_{r=1}^\infty(\kappa_r - \gamma_r)\frac{(-D)^r}{r!}\right]\Psi(x)\,.

If Ψ is chosen as the normal density with mean and variance as given by F, that is, mean \mu = \kappa_1 and variance \sigma^2 = \kappa_2, then the expansion becomes

F(x) = \exp\left[\sum_{r=3}^\infty\kappa_r\frac{(-D)^r}{r!}\right]\frac{1}{\sqrt{2\pi}\sigma}\exp\left[-\frac{(x-\mu)^2}{2\sigma^2}\right].

since \gamma_r=0 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. If we include only the first two correction terms to the normal distribution, we obtain

F(x) \approx \frac{1}{\sqrt{2\pi}\sigma}\exp\left[-\frac{(x-\mu)^2}{2\sigma^2}\right]\left[1+\frac{\kappa_3}{3!\sigma^3}H_3\left(\frac{x-\mu}{\sigma}\right)+\frac{\kappa_4}{4!\sigma^4}H_4\left(\frac{x-\mu}{\sigma}\right)\right]\,,

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 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.

The Edgeworth series

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.

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

Y_n = \frac{1}{\sqrt{n}} \sum_{i=1}^n \frac{X_i - \mu}{\sigma}.

Let Fn denote the cumulative distribution functions of the variables Yn. Then by the central limit theorem,

\lim_{n\to\infty} F_n(x) = \Phi(x) \equiv \int_{-\infty}^x \tfrac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}q^2}dq

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

Now assume that the random variables Xi have mean μ, variance σ2, and higher cumulants κrrλr. If we expand in terms of the standard normal distribution, that is, if we set


then the cumulant differences in the formal expression of the characteristic function fn(t) of Fn are

\kappa^{F(n)}_1-\gamma_1 = 0,
\kappa^{F(n)}_2-\gamma_2 = 0,
\kappa^{F(n)}_r-\gamma_r = \frac{\kappa_r}{\sigma^rn^{r/2-1}} = \frac{\lambda_r}{n^{r/2-1}}; \qquad r\geq 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}}\right] \exp(-t^2/2)\,,

where Pj(x) is a polynomial of degree 3j. Again, after inverse Fourier transform, the density function Fn follows as

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

The first five terms of the expansion are[5]

\begin{align} F_n(x) &= \Phi(x) \\ &\quad -\frac{1}{n^{\frac{1}{2}}}\left(\tfrac{1}{6}\lambda_3\,\Phi^{(3)}(x) \right) \\ &\quad +\frac{1}{n}\left(\tfrac{1}{24}\lambda_4\,\Phi^{(4)}(x) + \tfrac{1}{72}\lambda_3^2\,\Phi^{(6)}(x) \right) \\ &\quad -\frac{1}{n^{\frac{3}{2}}}\left(\tfrac{1}{120}\lambda_5\,\Phi^{(5)}(x) + \tfrac{1}{144}\lambda_3\lambda_4\,\Phi^{(7)}(x) + \tfrac{1}{1296}\lambda_3^3\,\Phi^{(9)}(x)\right) \\ &\quad + \frac{1}{n^2}\left(\tfrac{1}{720}\lambda_6\,\Phi^{(6)}(x) + \left(\tfrac{1}{1152}\lambda_4^2 + \tfrac{1}{720}\lambda_3\lambda_5\right)\Phi^{(8)}(x) + \tfrac{1}{1728}\lambda_3^2\lambda_4\,\Phi^{(10)}(x) + \tfrac{1}{31104}\lambda_3^4\,\Phi^{(12)}(x) \right)\\ &\quad + O \left (n^{-\frac{5}{2}} \right ). \end{align}

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 ϕ(n)(x)=(-1)nHn(x)ϕ(x), (where Hn 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.[6]

Illustration: density of the sample mean of 3 Χ²

Density of the sample mean of three chi2 variables. The chart compares the true density, the normal approximation, and two edgeworth expansions

Take X_i \sim \chi^2(k=2) \qquad i=1, 2, 3 and the sample mean \bar X = \frac{1}{3} \sum_{i=1}^{3} X_i .

We can use several distributions for \bar X :

  • The exact distribution, which follows a gamma distribution: \bar X \sim \mathrm{Gamma}\left(\alpha=n\cdot k /2, \theta= 2/n \right) = \mathrm{Gamma}\left(\alpha=3, \theta= 2/3 \right)
  • The asymptotic normal distribution: \bar X \xrightarrow{n \to \infty} N(k, 2\cdot k /n ) = N(2, 4/3 )
  • Two Edgeworth expansion, of degree 2 and 3

Disadvantages of the Edgeworth expansion

Edgeworth expansions can suffer from a few issues:

  • They are not guaranteed to be a proper probability distribution as:
    • The integral of the density needs not integrate to 1
    • Probabilities can be negative
  • They can be inaccurate, especially in the tails, due to mainly two reasons:
    • They are obtained under a Taylor series around the mean
    • They guarantee (asymptotically) an absolute error, not a relative one. This is an issue when one wants to approximate very small quantities, for which the absolute error might be small, but the relative error important.

See also


  1. ^ Stuart, A., & Kendall, M. G. (1968). The advanced theory of statistics. Hafner Publishing Company.
  2. ^ Kolassa, J. E. (2006). Series approximation methods in statistics (Vol. 88). Springer Science & Business Media.
  3. ^ Wallace, D. L. (1958). Asymptotic approximations to distributions. The Annals of Mathematical Statistics, 635-654.
  4. ^ Hall, P. (2013). The bootstrap and Edgeworth expansion. Springer Science & Business Media.
  5. ^ Weisstein, Eric W., "Edgeworth Series", MathWorld.
  6. ^ Kolassa, John E.; McCullagh, Peter (1990). "Edgeworth series for lattice distributions".  

Further reading

  • H. Cramér. (1957). Mathematical Methods of Statistics. Princeton University Press, Princeton.
  • D. L. Wallace. (1958). "Asymptotic approximations to distributions". Annals of Mathematical Statistics, 29: 635–654.
  • M. Kendall & A. Stuart. (1977), The advanced theory of statistics, Vol 1: Distribution theory, 4th Edition, Macmillan, New York.
  • 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.
  • Hazewinkel, Michiel, ed. (2001), "Edgeworth series",  
  • S. Blinnikov and R. Moessner (1998). Expansions for nearly Gaussian distributions. Astronomy and astrophysics Supplement series, 130: 193–205.
  • J. E. Kolassa (2006). Series Approximation Methods in Statistics (3rd ed.). (Lecture Notes in Statistics #88). Springer, New York.
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.