Expansions for nearly Gaussian distributions

By S. Blinnikov, R. Moessner. 1997

Various types of expansions in series of Cheby shev-Hermite polynomials currently used in astrophysics for weakly non-normal distributions are compared, namely the Gram-Charlier, Gauss-Hermite and Edgeworth expansions. It is shown that the Gram-Charlier series is most suspect because of its poor convergence properties. The Gauss-Hermite expansion is better but it has no intrinsic measure of accuracy. The best results are achieved with the asymptotic Edgeworth expansion. We draw attention to the form of this expansion found by Petrov for arbitrary order of the asymptotic parameter and present a simple algorithm realizing Petrov's prescription for the Edgeworth expansion. The results are illustrated by examples similar to the problems arising when fitting spectral line profiles of galaxies, supernovae, or other stars, and for the case of approximating the probability distribution of peculiar velocities in the cosmic string model of structure formation.

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Citations in the corpus

Continual Learning with Fully Probabilistic Models (2021, background): “This is trivial to show by expanding their distribution around a Gaussian one into an Edgeworth series (see [37]), and plugging this expansion into Eq.”
Probabilistic Power Flow Studies for Transmission Systems With Photovoltaic Generation Using Cumulants (2012, background): “Reference [20] gives an example of the distribution, in which the Gram-Charlier expansion is divergent while Edgeworth expansion gives a better result.”
Probabilistic Load Flow Modeling Comparing Maximum Entropy and Gram-Charlier Probability Density Function Reconstructions (2013, background): “space observation [23] and ocean wind behavior [30]).”
Probabilistic load flow in systems with wind generation (2009, background): “Numerical errors may make this convergence worse [24].”
Probabilistic load flow with wind production uncertainty using cumulants and Cornish–Fisher expansion (2009, background): “In many statistical inquiries we are more interested in the tails of a distribution than its behavior in the neighborhood of the mode, and it is here that the Gram Charlier series appears particularly inadequate [7], [9], [10].”
Causality detection based on information-theoretic approaches in time series analysis (2007, methodology): “They compared several methods for MI estimation, namely a standard (equally binned) histogram method (HG), an adaptive partitioning histogram method (AP) from Darbellay and Vajda [54] and the MI estimation based on the Gram–Charlier polynomial expansion (GC) [35].”
A Novel Probabilistic Optimal Power Flow Model With Uncertain Wind Power Generation Described by Customized Gaussian Mixture Model (2016, unknown): “Gram-Charlier (GC) [23], Edgeworth (EW) [24] and Cornish-Fish (CF) [25] have been used to approximate pdfs of RVs.”
Analytical solutions to sampling effects in drop size distribution measurements during stationary rainfall (2006, background): “Note, however, that the expansion for p(z) is asymptotic in the sense that the ratio between the series and p(z) converges to 1 as ns !1 . For finite ns the series may even diverge (e.g., Blinnikov and Moessner, 1998 ).”