Edgeworth expansions …

It’s interesting how the investigation of a given topic in statistics (in this example the topic will be the asymptotic behaviour of confidence intervals) can lead to the discovery that a key device for studying the said topic (in this example the device will be Edgeworth expansions) is also a key device for studying some unrelated statistical phenomenon (in this example the phenomenon will be non-Gaussianity in the cosmic microwave background).  In brief, Edgeworth expansions provide an asymptotic series approximation to a given distribution function in terms composed of polynomial combinations of cumulants multiplied by derivatives of the Normal distribution of increasing order.  For statistical purposes the Edgeworth series provides a powerful approximation to the distribution of a standardised sum of iid random variables, illuminating the rate at which it converges to the standard Normal as per the central limit theorem (e.g. Hall 1992).  While in cosmology the Edgeworth series provides a metric against which to evaluate or represent non-Gaussianity in the CMB (e.g. Juszkiewicz et al. 1995).

One of the first references I found online to help me get a handle on the Edgeworth series was the A&A paper “Expansions for nearly Gaussian distributions” by Blinnikov & Moessner (1998).  This paper has a nice description of the differences between three related expansions (Gram-Charlier, Gauss-Hermite and Edgeworth) and in particular a clear derivation of the Edgeworth expansion to arbitrary order (for a univariate random variable for which the density exists w.r.t. Lebesgue measure).  I say ‘clear’ but in fact it seems to me the authors’ proof falls over at the last hurdle.

In their Eqn 35 they define S_n = \kappa_n / \sigma^{2n-2} which allows the authors to write Eqn 34 in such a way that it looks like it contains a series of increasing positive powers of \sigma … after which they make a Taylor series of the exponential of this power series about \sigma=0.  This bothers me since the original characteristic function in Eqn 33 is ill-defined at \sigma = 0. It also seems they then claim that S_{r+2} \sigma^r = 0 when \sigma=0 (to get the 1 in the right hand side of Eqn 36), which is ignoring the dependence of S_{r+2} on \sigma from its definition, right? Likewise they treat S_n as no longer a function of \sigma when differentiating in Eqn 37.  None of this sits well with me.

Wouldn’t a correct derivation be to forget this expansion in terms of \sigma and just use directly the definition of the exponential (\exp(X) = 1 + X + X^2/2\mathrm{!} + \ldots) to expand Eqn 36 as a composition of power series from which the terms of each order can be grouped (the same method as used in their Appendix A)?

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2 Responses to Edgeworth expansions …

  1. telescoper says:

    The Edgeworth expansion can be found in many stand texts, even in Kendall & Stuart.

    The main problem with it as I see it is that it does not enforce positivity of the resulting PDF so is generally only meaningful for very slight departures from gaussianity.

    • It’s certainly unsatisfying to plot an expansion that takes on negative densities in places, but fortunately that’s not the direction the series is used for in the problem I was looking into (rather just the shape of the first few terms was of interest; e.g. Brown et al. 2002; https://projecteuclid.org/euclid.ss/1009213286 ).

      Until last week I genuinely hadn’t encountered the Edgeworth series since it’s not mentioned at all in Billingsley and only briefly mentioned in Feller, my two go-to references.

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