Probability distribution
Inprobability andstatistics theextended negative binomial distribution is adiscrete probability distribution extending thenegative binomial distribution . It is atruncated version of the negative binomial distribution[ 1] [ 2]
In the context ofactuarial science , the distribution appeared in its general form in a paper by K. Hess, A. Liewald and K.D. Schmidt[ 3] Panjer recursion works. For the casem = 1[ 4] logarithmic distribution and the negative binomial distribution by H.U. Gerber.[ 5]
Probability mass function [ edit ] For a natural numberm ≥ 1p ,r with0 <p ≤ 1 and–m <r < –m + 1 , theprobability mass function of the ExtNegBin(m , r , p ) distribution is given by
f ( k ; m , r , p ) = 0 for k ∈ { 0 , 1 , … , m − 1 } {\displaystyle f(k;m,r,p)=0\qquad {\text{ for }}k\in \{0,1,\ldots ,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 , {\displaystyle f(k;m,r,p)={\frac {{k+r-1 \choose k}p^{k}}{(1-p)^{-r}-\sum _{j=0}^{m-1}{j+r-1 \choose j}p^{j}}}\quad {\text{for }}k\in {\mathbb {N} }{\text{ with }}k\geq m,} where
( k + r − 1 k ) = Γ ( k + r ) k ! Γ ( r ) = ( − 1 ) k ( − r k ) ( 1 ) {\displaystyle {k+r-1 \choose k}={\frac {\Gamma (k+r)}{k!\,\Gamma (r)}}=(-1)^{k}\,{-r \choose k}\qquad \qquad (1)} is the (generalized)binomial coefficient andΓ denotes thegamma function .
Probability generating function [ edit ] Using thatf ( . ; m , r , ps )s ∈(0, 1] is also a probability mass function, it follows that theprobability 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 . {\displaystyle {\begin{aligned}\varphi (s)&=\sum _{k=m}^{\infty }f(k;m,r,p)s^{k}\\&={\frac {(1-ps)^{-r}-\sum _{j=0}^{m-1}{\binom {j+r-1}{j}}(ps)^{j}}{(1-p)^{-r}-\sum _{j=0}^{m-1}{\binom {j+r-1}{j}}p^{j}}}\qquad {\text{for }}|s|\leq {\frac {1}{p}}.\end{aligned}}} For the important casem = 1r ∈(–1, 0) , this simplifies to
φ ( s ) = 1 − ( 1 − p s ) − r 1 − ( 1 − p ) − r for | s | ≤ 1 p . {\displaystyle \varphi (s)={\frac {1-(1-ps)^{-r}}{1-(1-p)^{-r}}}\qquad {\text{for }}|s|\leq {\frac {1}{p}}.} ^ Jonhnson, N.L.; Kotz, S.;Kemp, A.W. (1993)Univariate Discrete Distributions , 2nd edition, WileyISBN 0-471-54897-9 ^ Shah S.M. (1971) "The displaced negative binomial distribution",Calcutta Statistical Association Bulletin ^ Hess, Klaus Th.; Anett Liewald; Klaus D. Schmidt (2002)."An extension of Panjer's recursion" (PDF) .ASTIN Bulletin .32 (2):283– 297.doi :10.2143/AST.32.2.1030 MR 1942940 .Zbl 1098.91540 . ^ Willmot, Gordon (1988)."Sundt and Jewell's family of discrete distributions" (PDF) .ASTIN Bulletin .18 (1):17– 29.doi :10.2143/AST.18.1.2014957 ^ Gerber, Hans U. (1992). "From the generalized gamma to the generalized negative binomial distribution".Insurance: Mathematics and Economics .10 (4):303– 309.doi :10.1016/0167-6687(92)90061-F .ISSN 0167-6687 .MR 1172687 .Zbl 0743.62014 .
Discrete
with finite with infinite
Continuous
supported on a supported on a supported with support
Mixed
Multivariate Directional Degenerate singular Families
