Expansions for nearly Gaussian distributions

¹ Institute for Theoretical and Experimental Physics, 117259, Moscow, Russia Sternberg Astronomical Institute, 119899 Moscow, Russia
² Max-Planck-Institut für Astrophysik, D-85740 Garching, Germany

Send offprint request to: S. Blinnikov

Received: 8 July 1997
Accepted: 10 November 1997

Abstract

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.