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The Early History of the Cumulants and the Gram‐Charlier Series
Author(s) -
Hald Anders
Publication year - 2000
Publication title -
international statistical review
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.051
H-Index - 54
eISSN - 1751-5823
pISSN - 0306-7734
DOI - 10.1111/j.1751-5823.2000.tb00318.x
Subject(s) - cumulant , mathematics , edgeworth series , hermite polynomials , taylor series , generalization , series (stratigraphy) , central limit theorem , moment (physics) , mathematical analysis , laplace transform , statistics , paleontology , physics , classical mechanics , biology
Summary The early history of the Gram‐Charlier series is discussed from three points of view: (1) a generalization of Laplace's central limit theorem, (2) a least squares approximation to a continuous function by means of Chebyshev‐Hermite polynomials, (3) a generalization of Gauss's normal distribution to a system of skew distributions. Thiele defined the cumulants in terms of the moments, first by a recursion formula and later by an expansion of the logarithm of the moment generating function. He devised a differential operator which adjusts any cumulant to a desired value. His little known 1899 paper in Danish on the properties of the cumulants is translated into English in the Appendix.