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, 3, ... } PMF
Γ
(
r
+
k
)
k
!
Γ
(
r
)
B
(
α
+
r
,
β
+
k
)
B
(
α
,
β
)
{\displaystyle {\frac {\Gamma (r+k)}{k!\;\Gamma (r)}}{\frac {\mathrm {B} (\alpha +r,\beta +k)}{\mathrm {B} (\alpha ,\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
(
α
+
r
−
1
)
β
(
α
+
β
−
1
)
(
α
−
2
)
(
α
−
1
)
2
if
α
>
2
∞
otherwise
{\displaystyle {\begin{cases}{\frac {r(\alpha +r-1)\beta (\alpha +\beta -1)}{(\alpha -2){(\alpha -1)}^{2}}}&{\text{if}}\ \alpha >2\\\infty &{\text{otherwise}}\ \end{cases}}}
Skewness
{
(
α
+
2
r
−
1
)
(
α
+
2
β
−
1
)
(
α
−
3
)
r
(
α
+
r
−
1
)
β
(
α
+
β
−
1
)
α
−
2
if
α
>
3
∞
otherwise
{\displaystyle {\begin{cases}{\frac {(\alpha +2r-1)(\alpha +2\beta -1)}{(\alpha -3){\sqrt {\frac {r(\alpha +r-1)\beta (\alpha +\beta -1)}{\alpha -2}}}}}&{\text{if}}\ \alpha >3\\\infty &{\text{otherwise}}\ \end{cases}}}
MGF
undefined CF
B
(
α
,
β
+
r
)
B
(
α
,
β
)
2
F
1
(
r
,
α
;
α
+
β
+
r
;
e
i
t
)
{\displaystyle {\frac {\mathrm {B} (\alpha ,\beta +r)}{\mathrm {B} (\alpha ,\beta )}}{}_{2}F_{1}(r,\alpha ;\alpha +\beta +r;e^{it})\!}
beta function and 2 F1 is the hypergeometric function .
In probability theory , a beta negative binomial distribution is the probability distribution of a discrete random variable X equal to the number of failures needed to get r successes in a sequence of independent Bernoulli trials where the probability p of success on each trial is constant within any given experiment but is itself a random variable following a beta distribution , varying between different experiments. 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] beta-Pascal distribution .[ 1]
If parameters of the beta distribution are α and β , and if
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 is a beta negative binomial distribution:
X
∼
B
N
B
(
r
,
α
,
β
)
.
{\displaystyle X\sim \mathrm {BNB} (r,\alpha ,\beta ).}
In the above, NB(r , p ) is the negative binomial distribution and B(α , β ) is the beta distribution .
Definition
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
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
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
)
(
r
+
α
+
β
)
(
k
)
{\displaystyle f(k|\alpha ,\beta ,r)={\frac {r^{(k)}\alpha ^{(r)}\beta ^{(k)}}{k!(\alpha +\beta )^{(r)}(r+\alpha +\beta )^{(k)}}}}
Properties
non-identifiable
The beta negative binomial is Identifiability which can be seen easily by simply swapping $r$ and $\beta$ in the above density and noting that the density is unchanged.
=== Relation to other distributions
The beta negative binomial distribution contains the beta geometric distribution as a special case when
r
=
1
{\displaystyle r=1}
geometric distribution arbitrarily well. It also approximates the negative binomial distribution arbitrary well for large
α
{\displaystyle \alpha }
β
{\displaystyle \beta }
Poisson distribution arbitrarily well for large
α
{\displaystyle \alpha }
β
{\displaystyle \beta }
r
{\displaystyle r}
=== Heavy tailed
By Stirling's approximation to the beta function, it can be easily shown that
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 .
See also
Notes
^ a b Johnson et al. (1993)
References
Jonhnson, 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
External links
Discrete
with finite with infinite
Continuous
supported on a supported on a supported with support
Mixed
Multivariate Directional Degenerate singular Families
