From Wikipedia, the free encyclopedia
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 = 1 , 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 ≥ 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
}
{\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 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
.
{\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 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
.
{\displaystyle \varphi (s)={\frac {1-(1-ps)^{-r}}{1-(1-p)^{-r}}}\qquad {\text{for }}|s|\leq {\frac {1}{p}}.}
References
^ Jonhnson, N.L.; Kotz, S.; Kemp, A.W. (1993) Univariate Discrete Distributions , 2nd edition, Wiley
ISBN
0-471-54897-9 (page 227)
^ Shah S.M. (1971) "The displaced negative binomial distribution", Bulletin of the Calcutta Statistical Association , 20, 143–152
^ 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 univariate
with finite support with infinite support
Continuous univariate
supported on a bounded interval supported on a semi-infinite interval supported on the whole real line with support whose type varies
Mixed univariate
Multivariate (joint)
Directional
Degenerate and
singular Families