- Edgeworth series
The Gram-Charlier A series and the Edgeworth series, named in honor of
Francis Ysidro Edgeworth, are series that approximate a probability distributionin 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 functionof the distribution whose probability density functionis "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 , characteristic function , and standardized 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")"r"ψ("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
If 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 "H"3("x") = "x"3 − 3"x" and "H"4("x") = "x"4 − 6"x"2 + 3 (these are
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 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
Further, let "F""n" be the probability density function of the variables "Y""n". 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. (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
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
polynomialof degree 3"j". Again, after inverse Fourier transform, the density function "F""n" follows as
The first three terms of the expansion are (Cramér 1957)
Here, ("j")("x") is the "j"th derivative of ("x") with respect to "x". Blinnikov and Moessner (1998) have given a simple algorithm to calculate higher-order terms of the expansion.
* 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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