Expansions for nearly Gaussian distributions

Details Expansions for nearly Gaussian distributions Expansions for nearly Gaussian distributions

1997-11-20

arxiv.org/abs/astro-ph/9711239v1

Astron.Astrophys.Suppl.Ser. 130 (1998) 193-205

Astronomy and Astrophysics

Astrophysics

13 pages with 11 eps figures, aa.cls + graphics packages, Submitted to Astronomy & Astrophysics Supplement on July 16, 1997, a

Scientific paper

10.1051/aas:1998221

Various types of expansions in series of Chebyshev-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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