Far East Journal of Theoretical Statistics
Volume 7, Issue 1, Pages 1 - 17
(May 2002)
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THE GEOMETRIC DISTRIBUTION’S CENTRAL MOMENTS AND EULERIAN
NUMBERS OF THE SECOND KIND
L. R. Shenton (U. S. A) and K. O. Bowman (U. S. A)
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Abstract: In a previous paper [Far East J. Theo. Stat. 5(1) (2001),
113-142], we have studied cumulants of the geometric distribution, showing that
they lead to Eulerian numbers such as the sets
These Eulerian numbers (of the first kind) relate to partitions of s from
the factorial s!, and as discrete distributions which appear to be
nearly the normal distribution when s is large.
Eulerian numbers of the 2nd kind are from the central moments
of the geometric distribution and contain sets such as
These are symmetric partitions of where
refers to the truncated series for
the negative exponential
The basic results spring from a finite difference equation of the first
order, this being solved by usage of the finite difference operator
As with Eulerian numbers of the 1st kind, there is the
property of normality; actually the Eulerian numbers of the 2nd kind, viewed as
discrete distribution, are asymptotically normally distributed. |
Keywords and phrases: asymptotic moments, asymptotic series, difference-differential equation,
maximum likelihood estimator, power series distributions, recurrence relations. |
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