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

1 Gram–Charlier A series
2 Edgeworth series
3 References
4 Further reading

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 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:

$f(t)=\exp\left[\sum_{r=1}^\infty(\kappa_r-\gamma_r)\frac{(it)^r}{r!}\right]\psi(t)\,.$

By the properties of the Fourier transform, (it)^rψ(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

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

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

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

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

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

with H₃(x) = x³ − 3x and H₄(x) = x⁴ − 6x² + 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(−x²/4) 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 and identically distributed random variables with mean μ and variance σ², and let Y_n be their standardized sums:

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

Let F_n denote the cumulative distribution functions of the variables Y_n. 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 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

$\Psi(x)=\frac{1}{\sqrt{2\pi}}\exp(-\tfrac{1}{2}x^2)$

then the cumulant differences in the formal expression of the characteristic function f_n(t) of F_n are

$\kappa^{F(n)}_1-\gamma_1 = 0\,,$

$\kappa^{F(n)}_2-\gamma_2 = 0\,,$

$\kappa^{F(n)}_r-\gamma_r = \frac{\kappa_r}{\sigma^rn^{r/2-1}} = \frac{\lambda_r}{n^{r/2-1}}; \quad r\geq 3\,.$

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

$f_n(t)=\left[1%2B\sum_{j=1}^\infty \frac{P_j(it)}{n^{j/2}}\right] \exp(-t^2/2)\,,$

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

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

The first five terms of the expansion are^[1]

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

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.

References

^ Weisstein, Eric W., "Edgeworth Series" from MathWorld.

Edgeworth series

Contents

Gram–Charlier A series

Edgeworth series

References

Further reading