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  4. 2026
  5. VOL 48, NO 2 (2026)
  6. pp. 03-23

The Excess Coefficient Analysis of Kaplansky's Generalized Distributions

O.I. Krasilnikov
ORCID: https://orcid.org/0000-0001-5666-6459https://orcid.org/0000-0001-5666-6459

Èlektron. model. 2026, 48(2):03-23

ABSTRACT

The probability densities I ( ) p x , …, IV( ) p x , given in Kaplansky's report (1945), as examples refuting the interpretation of the excess coefficient as a measure of the sharpness of distributions are analyzed. Kaplan's generalized distributions K1, ..., K4 are defined, whose probability densities I ( ; ) p x p , …, IV( ; ) p x p are a group of two-component mixtures of sym metric distributions with an arbitrary weight coefficient. The dependence of the properties of probability densities I ( ; ) p x p , …, IV( ; ) p x p and their excess coefficient 4( )  p on the weight coefficient p is investigated. A comparison is made between the zero values of the standar dized probability densities I ( ; ) p x p , …, IV( ; ) p x p and the value of the standard normal distri bution. The results obtained allow for mathematical and computer modeling of a multitude of examples of non-Gaussian symmetric probability densities, which refute the interpretation of the excess coefficient as a measure of the sharpness of distributions.

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KEYWORDS

non-Gaussian symmetric distributions, excess kurtosis, two-component mixtures of distributions, cumulant coefficients, cumulant analysis.

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