Beta Negative Binomial Parameters
α
>
0
{\displaystyle \alpha >0}
shape (real )
β
>
0
{\displaystyle \beta >0}
shape (real )
r
>
0
{\displaystyle r>0}
integer but can be extended to real ) Support
k
∈
{
0
,
1
,
2
,
…
}
{\displaystyle k\in \{0,1,2,\ldots \}}
PMF
B
(
r
+
k
,
α
+
β
)
B
(
r
,
α
)
Γ
(
k
+
β
)
k
!
Γ
(
β
)
{\displaystyle {\frac {\mathrm {B} (r+k,\alpha +\beta )}{\mathrm {B} (r,\alpha )}}{\frac {\Gamma (k+\beta )}{k!\;\Gamma (\beta )}}}
Mean
{
r
β
α
−
1
if
α
>
1
∞
otherwise
{\displaystyle {\begin{cases}{\frac {r\beta }{\alpha -1}}&{\text{if}}\ \alpha >1\\\infty &{\text{otherwise}}\ \end{cases}}}
Variance
{
r
β
(
r
+
α
−
1
)
(
β
+
α
−
1
)
(
α
−
2
)
(
α
−
1
)
2
if
α
>
2
∞
otherwise
{\displaystyle {\begin{cases}{\frac {r\beta (r+\alpha -1)(\beta +\alpha -1)}{(\alpha -2){(\alpha -1)}^{2}}}&{\text{if}}\ \alpha >2\\\infty &{\text{otherwise}}\ \end{cases}}}
Skewness
{
(
2
r
+
α
−
1
)
(
2
β
+
α
−
1
)
(
α
−
3
)
r
β
(
r
+
α
−
1
)
(
β
+
α
−
1
)
α
−
2
if
α
>
3
∞
otherwise
{\displaystyle {\begin{cases}{\frac {(2r+\alpha -1)(2\beta +\alpha -1)}{(\alpha -3){\sqrt {\frac {r\beta (r+\alpha -1)(\beta +\alpha -1)}{\alpha -2}}}}}&{\text{if}}\ \alpha >3\\\infty &{\text{otherwise}}\ \end{cases}}}
MGF
does not exist CF
2
F
1
(
β
,
r
;
α
+
β
+
r
;
e
i
t
)
(
α
)
(
r
)
(
α
+
β
)
(
r
)
{\displaystyle {}_{2}F_{1}(\beta ,r;\alpha +\beta +r;e^{it}){\frac {(\alpha )^{(r)}}{(\alpha +\beta )^{(r)}}}\!}
(
x
)
(
r
)
=
Γ
(
x
+
r
)
Γ
(
x
)
{\displaystyle (x)^{(r)}={\frac {\Gamma (x+r)}{\Gamma (x)}}}
Pochhammer symbol and
2
F
1
{\displaystyle {}_{2}F_{1}}
hypergeometric function . PGF
2
F
1
(
β
,
r
;
α
+
β
+
r
;
z
)
(
α
)
(
r
)
(
α
+
β
)
(
r
)
{\displaystyle {}_{2}F_{1}(\beta ,r;\alpha +\beta +r;z){\frac {(\alpha )^{(r)}}{(\alpha +\beta )^{(r)}}}}
In probability theory , a beta negative binomial distribution is the probability distribution of a discrete random variable
X
{\displaystyle X}
r
{\displaystyle r}
independent Bernoulli trials . The probability
p
{\displaystyle p}
beta distribution . Thus the distribution is a compound probability distribution .
This distribution has also been called both the inverse Markov-Pólya distribution and the generalized Waring distribution [ 1] BNB distribution. A shifted form of the distribution has been called the beta-Pascal distribution .[ 1]
If parameters of the beta distribution are
α
{\displaystyle \alpha }
β
{\displaystyle \beta }
X
∣
p
∼
N
B
(
r
,
p
)
,
{\displaystyle X\mid p\sim \mathrm {NB} (r,p),}
where
p
∼
B
(
α
,
β
)
,
{\displaystyle p\sim {\textrm {B}}(\alpha ,\beta ),}
then the marginal distribution of
X
{\displaystyle X}
posterior predictive distribution ) is a beta negative binomial distribution:
X
∼
B
N
B
(
r
,
α
,
β
)
.
{\displaystyle X\sim \mathrm {BNB} (r,\alpha ,\beta ).}
In the above,
N
B
(
r
,
p
)
{\displaystyle \mathrm {NB} (r,p)}
negative binomial distribution and
B
(
α
,
β
)
{\displaystyle {\textrm {B}}(\alpha ,\beta )}
beta distribution .
Definition and derivation [ edit ] Denoting
f
X
|
p
(
k
|
q
)
,
f
p
(
q
|
α
,
β
)
{\displaystyle f_{X|p}(k|q),f_{p}(q|\alpha ,\beta )}
f
(
k
|
α
,
β
,
r
)
{\displaystyle f(k|\alpha ,\beta ,r)}
f
(
k
|
α
,
β
,
r
)
=
∫
0
1
f
X
|
p
(
k
|
r
,
q
)
⋅
f
p
(
q
|
α
,
β
)
d
q
=
∫
0
1
(
k
+
r
−
1
k
)
(
1
−
q
)
k
q
r
⋅
q
α
−
1
(
1
−
q
)
β
−
1
B
(
α
,
β
)
d
q
=
1
B
(
α
,
β
)
(
k
+
r
−
1
k
)
∫
0
1
q
α
+
r
−
1
(
1
−
q
)
β
+
k
−
1
d
q
{\displaystyle {\begin{aligned}f(k|\alpha ,\beta ,r)\;=&\;\int _{0}^{1}f_{X|p}(k|r,q)\cdot f_{p}(q|\alpha ,\beta )\mathrm {d} q\\=&\;\int _{0}^{1}{\binom {k+r-1}{k}}(1-q)^{k}q^{r}\cdot {\frac {q^{\alpha -1}(1-q)^{\beta -1}}{\mathrm {B} (\alpha ,\beta )}}\mathrm {d} q\\=&\;{\frac {1}{\mathrm {B} (\alpha ,\beta )}}{\binom {k+r-1}{k}}\int _{0}^{1}q^{\alpha +r-1}(1-q)^{\beta +k-1}\mathrm {d} q\end{aligned}}}
Noting that the integral evaluates to:
∫
0
1
q
α
+
r
−
1
(
1
−
q
)
β
+
k
−
1
d
q
=
Γ
(
α
+
r
)
Γ
(
β
+
k
)
Γ
(
α
+
β
+
k
+
r
)
{\displaystyle \int _{0}^{1}q^{\alpha +r-1}(1-q)^{\beta +k-1}\mathrm {d} q={\frac {\Gamma (\alpha +r)\Gamma (\beta +k)}{\Gamma (\alpha +\beta +k+r)}}}
we can arrive at the following formulas by relatively simple manipulations.
If
r
{\displaystyle r}
beta function ,:
f
(
k
|
α
,
β
,
r
)
=
(
r
+
k
−
1
k
)
B
(
α
+
r
,
β
+
k
)
B
(
α
,
β
)
{\displaystyle f(k|\alpha ,\beta ,r)={\binom {r+k-1}{k}}{\frac {\mathrm {B} (\alpha +r,\beta +k)}{\mathrm {B} (\alpha ,\beta )}}}
More generally, the PMF can be written
f
(
k
|
α
,
β
,
r
)
=
Γ
(
r
+
k
)
k
!
Γ
(
r
)
B
(
α
+
r
,
β
+
k
)
B
(
α
,
β
)
{\displaystyle f(k|\alpha ,\beta ,r)={\frac {\Gamma (r+k)}{k!\;\Gamma (r)}}{\frac {\mathrm {B} (\alpha +r,\beta +k)}{\mathrm {B} (\alpha ,\beta )}}}
or
f
(
k
|
α
,
β
,
r
)
=
B
(
r
+
k
,
α
+
β
)
B
(
r
,
α
)
Γ
(
k
+
β
)
k
!
Γ
(
β
)
{\displaystyle f(k|\alpha ,\beta ,r)={\frac {\mathrm {B} (r+k,\alpha +\beta )}{\mathrm {B} (r,\alpha )}}{\frac {\Gamma (k+\beta )}{k!\;\Gamma (\beta )}}}
PMF expressed with Gamma [ edit ] Using the properties of the Beta function , the PMF with integer
r
{\displaystyle r}
f
(
k
|
α
,
β
,
r
)
=
(
r
+
k
−
1
k
)
Γ
(
α
+
r
)
Γ
(
β
+
k
)
Γ
(
α
+
β
)
Γ
(
α
+
r
+
β
+
k
)
Γ
(
α
)
Γ
(
β
)
{\displaystyle f(k|\alpha ,\beta ,r)={\binom {r+k-1}{k}}{\frac {\Gamma (\alpha +r)\Gamma (\beta +k)\Gamma (\alpha +\beta )}{\Gamma (\alpha +r+\beta +k)\Gamma (\alpha )\Gamma (\beta )}}}
More generally, the PMF can be written as
f
(
k
|
α
,
β
,
r
)
=
Γ
(
r
+
k
)
k
!
Γ
(
r
)
Γ
(
α
+
r
)
Γ
(
β
+
k
)
Γ
(
α
+
β
)
Γ
(
α
+
r
+
β
+
k
)
Γ
(
α
)
Γ
(
β
)
{\displaystyle f(k|\alpha ,\beta ,r)={\frac {\Gamma (r+k)}{k!\;\Gamma (r)}}{\frac {\Gamma (\alpha +r)\Gamma (\beta +k)\Gamma (\alpha +\beta )}{\Gamma (\alpha +r+\beta +k)\Gamma (\alpha )\Gamma (\beta )}}}
PMF expressed with the rising Pochammer symbol [ edit ] The PMF is often also presented in terms of the Pochammer symbol for integer
r
{\displaystyle r}
f
(
k
|
α
,
β
,
r
)
=
r
(
k
)
α
(
r
)
β
(
k
)
k
!
(
α
+
β
)
(
r
+
k
)
{\displaystyle f(k|\alpha ,\beta ,r)={\frac {r^{(k)}\alpha ^{(r)}\beta ^{(k)}}{k!(\alpha +\beta )^{(r+k)}}}}
The k factorial moment of a beta negative binomial random variable X
k
<
α
{\displaystyle k<\alpha }
E
[
(
X
)
k
]
=
Γ
(
r
+
k
)
Γ
(
r
)
Γ
(
β
+
k
)
Γ
(
β
)
Γ
(
α
−
k
)
Γ
(
α
)
.
{\displaystyle \operatorname {E} {\bigl [}(X)_{k}{\bigr ]}={\frac {\Gamma (r+k)}{\Gamma (r)}}{\frac {\Gamma (\beta +k)}{\Gamma (\beta )}}{\frac {\Gamma (\alpha -k)}{\Gamma (\alpha )}}.}
The beta negative binomial is non-identifiable which can be seen easily by simply swapping
r
{\displaystyle r}
β
{\displaystyle \beta }
characteristic function and noting that it is unchanged. Thus estimation demands that a constraint be placed on
r
{\displaystyle r}
β
{\displaystyle \beta }
Relation to other distributions [ edit ] The beta negative binomial distribution contains the beta geometric distribution as a special case when either
r
=
1
{\displaystyle r=1}
β
=
1
{\displaystyle \beta =1}
geometric distribution arbitrarily well. It also approximates the negative binomial distribution arbitrary well for large
α
{\displaystyle \alpha }
Poisson distribution arbitrarily well for large
α
{\displaystyle \alpha }
β
{\displaystyle \beta }
r
{\displaystyle r}
By Stirling's approximation to the beta function, it can be easily shown that for large
k
{\displaystyle k}
f
(
k
|
α
,
β
,
r
)
∼
Γ
(
α
+
r
)
Γ
(
r
)
B
(
α
,
β
)
k
r
−
1
(
β
+
k
)
r
+
α
{\displaystyle f(k|\alpha ,\beta ,r)\sim {\frac {\Gamma (\alpha +r)}{\Gamma (r)\mathrm {B} (\alpha ,\beta )}}{\frac {k^{r-1}}{(\beta +k)^{r+\alpha }}}}
which implies that the beta negative binomial distribution is heavy tailed and that moments less than or equal to
α
{\displaystyle \alpha }
Beta geometric distribution [ edit ] The beta geometric distribution is an important special case of the beta negative binomial distribution occurring for
r
=
1
{\displaystyle r=1}
f
(
k
|
α
,
β
)
=
B
(
α
+
1
,
β
+
k
)
B
(
α
,
β
)
{\displaystyle f(k|\alpha ,\beta )={\frac {\mathrm {B} (\alpha +1,\beta +k)}{\mathrm {B} (\alpha ,\beta )}}}
This distribution is used in some Buy Till you Die (BTYD) models.
Further, when
β
=
1
{\displaystyle \beta =1}
Yule–Simon distribution . However, it is more common to define the Yule-Simon distribution in terms of a shifted version of the beta geometric. In particular, if
X
∼
B
G
(
α
,
1
)
{\displaystyle X\sim BG(\alpha ,1)}
X
+
1
∼
Y
S
(
α
)
{\displaystyle X+1\sim YS(\alpha )}
[ edit ] In the case when the 3 parameters
r
,
α
{\displaystyle r,\alpha }
β
{\displaystyle \beta }
urn model - or more specifically a basic Pólya urn model . Consider an urn initially containing
α
{\displaystyle \alpha }
β
{\displaystyle \beta }
r
{\displaystyle r}
X
{\displaystyle X}
B
N
B
(
r
,
α
,
β
)
{\displaystyle \mathrm {BNB} (r,\alpha ,\beta )}
r
+
α
{\displaystyle r+\alpha }
X
+
β
{\displaystyle X+\beta }
By the non-identifiability property,
X
{\displaystyle X}
α
{\displaystyle \alpha }
r
{\displaystyle r}
β
{\displaystyle \beta }
^ a b Johnson et al. (1993)
Johnson, N.L.; Kotz, S.; Kemp, A.W. (1993) Univariate Discrete Distributions , 2nd edition, Wiley ISBN 0-471-54897-9 (Section 6.2.3)
Kemp, C.D.; Kemp, A.W. (1956) "Generalized hypergeometric distributions, Journal of the Royal Statistical Society , Series B, 18, 202–211
Wang, Zhaoliang (2011) "One mixed negative binomial distribution with application", Journal of Statistical Planning and Inference , 141 (3), 1153-1160 doi :10.1016/j.jspi.2010.09.020
Discrete
with finite with infinite
Continuous
supported on a supported on a supported with support
Mixed
Multivariate Directional Degenerate singular Families
![{\displaystyle \operatorname {E} {\bigl [}(X)_{k}{\bigr ]}={\frac {\Gamma (r+k)}{\Gamma (r)}}{\frac {\Gamma (\beta +k)}{\Gamma (\beta )}}{\frac {\Gamma (\alpha -k)}{\Gamma (\alpha )}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38d2da3ee23b85c596422287cf0c382298b12318)
