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Extended negative binomial distribution

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In probability and statistics the extended negative binomial distribution is a discrete probability distribution extending the negative binomial distribution. It is a truncated version of the negative binomial distribution for which estimation methods have been studied.

Contents

In the context of actuarial science, the distribution appeared in its general form in a paper by K. Hess, A. Liewald and K.D. Schmidt when they characterized all distributions for which the extended Panjer recursion works. For the case m = 1, the distribution was already discussed by Willmot and put into a parametrized family with the logarithmic distribution and the negative binomial distribution by H.U. Gerber.

Probability mass function

For a natural number m ≥ 1 and real parameters p, r with 0 < p ≤ 1 and –m < r < –m + 1, the probability mass function of the ExtNegBin(m, r, p) distribution is given by

f ( k ; m , r , p ) = 0  for  k ∈ { 0 , 1 , … , m − 1 }

and

f ( k ; m , r , p ) = ( k + r − 1 k ) p k ( 1 − p ) − r − ∑ j = 0 m − 1 ( j + r − 1 j ) p j for  k ∈ N  with  k ≥ m ,

where

( k + r − 1 k ) = Γ ( k + r ) k ! Γ ( r ) = ( − 1 ) k ( − r k ) ( 1 )

is the (generalized) binomial coefficient and Γ denotes the gamma function.

Probability generating function

Using that f ( . ; m, r, ps) for s ∈ (0, 1] is also a probability mass function, it follows that the probability generating function is given by

φ ( s ) = ∑ k = m ∞ f ( k ; m , r , p ) s k = ( 1 − p s ) − r − ∑ j = 0 m − 1 ( j + r − 1 j ) ( p s ) j ( 1 − p ) − r − ∑ j = 0 m − 1 ( j + r − 1 j ) p j for  | s | ≤ 1 p .

For the important case m = 1, hence r ∈ (–1, 0), this simplifies to

φ ( s ) = 1 − ( 1 − p s ) − r 1 − ( 1 − p ) − r for  | s | ≤ 1 p .

References

Extended negative binomial distribution Wikipedia
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