Extended negative binomial distribution

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Template:Short description 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[1] for which estimation methods have been studied.[2]

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[3] when they characterized all distributions for which the extended Panjer recursion works. For the case m = 1Script error: No such module "Check for unknown parameters"., the distribution was already discussed by Willmot[4] and put into a parametrized family with the logarithmic distribution and the negative binomial distribution by H.U. Gerber.[5]

Probability mass function

For a natural number m ≥ 1Script error: No such module "Check for unknown parameters". and real parameters Template:Mvar, Template:Mvar with 0 < p ≤ 1Script error: No such module "Check for unknown parameters". and –m < r < –m + 1Script error: No such module "Check for unknown parameters"., the probability mass function of the ExtNegBin(Template:Mvar, Template:Mvar, Template:Mvar) distribution is given by

f(k;m,r,p)=0 for k∈{0,1,…,m−1}

and

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

where

(k+r−1k)=Γ(k+r)k!Γ(r)=(−1)k(−rk)(1)

is the (generalized) binomial coefficient and ΓScript error: No such module "Check for unknown parameters". denotes the gamma function.

Probability generating function

Using that f ( . ; m, r, ps)Script error: No such module "Check for unknown parameters". for s ∈ Script error: No such module "Check for unknown parameters".Template:Open-closed 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)sk=(1−ps)−r−∑j=0m−1(j+r−1j)(ps)j(1−p)−r−∑j=0m−1(j+r−1j)pjfor |s|≤1p.

For the important case m = 1Script error: No such module "Check for unknown parameters"., hence r ∈ Script error: No such module "Check for unknown parameters".Template:Open-open, this simplifies to

φ(s)=1−(1−ps)−r1−(1−p)−rfor |s|≤1p.

References

  1. ↑ Jonhnson, N.L.; Kotz, S.; Kemp, A.W. (1993) Univariate Discrete Distributions, 2nd edition, Wiley Template:ISBN (page 227)
  2. ↑ Shah S.M. (1971) "The displaced negative binomial distribution", Bulletin of the Calcutta Statistical Association, 20, 143–152
  3. ↑ Script error: No such module "Citation/CS1".
  4. ↑ Script error: No such module "Citation/CS1".
  5. ↑ Script error: No such module "Citation/CS1".

Template:ProbDistributions

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