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ⓘ Combinant. In the mathematical theory of probability, the combinants c n of a random variable X are defined via the combinant-generating function G, which is de ..




                                     

ⓘ Combinant

In the mathematical theory of probability, the combinants c n of a random variable X are defined via the combinant-generating function G, which is defined from the moment generating function M as

G X t = M X log ⁡ 1 + t) {\displaystyle G_{X}t=M_{X}\log1+t)}

which can be expressed directly in terms of a random variable X as

G X t:= E,\quad t\in \mathbb {R},}

wherever this expectation exists.

The n th combinant can be obtained as the n th derivatives of the logarithm of combinant generating function evaluated at –1 divided by n factorial:

c n = 1 n! ∂ n ∂ t n log ⁡ G t) | t = − 1 {\displaystyle c_{n}={\frac {1}{n!}}{\frac {\partial ^{n}}{\partial t^{n}}}\logGt){\bigg |}_{t=-1}}

Important features in common with the cumulants are:

  • the combinants share the additivity property of the cumulants;
  • for infinite divisibility probability distributions, both sets of moments are strictly positive.
                                     
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  • deviation skewness kurtosis L - moment moment - generating function mgf characteristic function probability - generating function pgf cumulant combinant

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