Neha Patil (Editor)

Beta negative binomial distribution

Updated on
Edit
Like
Comment
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit
Parameters
  
α > 0 {displaystyle alpha >0} shape (real) β > 0 {displaystyle eta >0} shape (real) r > 0 {displaystyle r>0} — number of failures until the experiment is stopped (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 { rac {Gamma (r+k)}{k!;Gamma (r)}}{ rac {mathrm {B} (alpha +r,eta +k)}{mathrm {B} (alpha ,eta )}}}

Mean
  
{ r β α − 1 if   α > 1 ∞ otherwise   {displaystyle {egin{cases}{ rac {reta }{alpha -1}}&{ ext{if}} alpha >1infty &{ ext{otherwise}} end{cases}}}

Variance
  
{ r ( α + r − 1 ) β ( α + β − 1 ) ( α − 2 ) ( α − 1 ) 2 if   α > 2 ∞ otherwise   {displaystyle {egin{cases}{ rac {r(alpha +r-1)eta (alpha +eta -1)}{(alpha -2){(alpha -1)}^{2}}}&{ ext{if}} alpha >2infty &{ ext{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 {egin{cases}{ rac {(alpha +2r-1)(alpha +2eta -1)}{(alpha -3){sqrt { rac {r(alpha +r-1)eta (alpha +eta -1)}{alpha -2}}}}}&{ ext{if}} alpha >3infty &{ ext{otherwise}} end{cases}}}

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.

Contents

This distribution has also been called both the inverse Markov-Pólya distribution and the generalized Waring distribution. A shifted form of the distribution has been called the beta-Pascal distribution.

If parameters of the beta distribution are α and β, and if

X ∣ p ∼ N B ( r , p ) ,

where

p ∼ B ( α , β ) ,

then the marginal distribution of X is a beta negative binomial distribution:

X ∼ B N B ( r , α , β ) .

In the above, NB(r, p) is the negative binomial distribution and B(α, β) is the beta distribution.

Recurrence relation

{ ( k + 1 ) p ( k + 1 ) ( α + β + k + r ) + ( β + k ) ( − k − r ) p ( k ) = 0 , p ( 0 ) = ( α ) r ( α + β ) r }

Definition

If r is an integer, then the PMF can be written in terms of the beta function,:

f ( k | α , β , r ) = ( r + k − 1 k ) B ( α + r , β + k ) B ( α , β ) .

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 ) .

PMF expressed with Gamma

Using the properties of the Beta function, the PMF with integer r can be rewritten as:

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 ) .

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 ) .

PMF expressed with the rising Pochammer symbol

The PMF is often also presented in terms of the Pochammer symbol for integer r

f ( k | α , β , r ) = r ( k ) α ( r ) β ( k ) k ! ( α + β ) ( r ) ( r + α + β ) ( k )

Properties

The beta negative binomial distribution contains the beta geometric distribution as a special case when r = 1 . It can therefore approximate the geometric distribution arbitrarily well. It also approximates the negative binomial distribution arbitrary well for large α and β . It can therefore approximate the Poisson distribution arbitrarily well for large α , β and r .

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

which implies that the beta negative binomial distribution is heavy tailed.

References

Beta negative binomial distribution Wikipedia
(Text) CC BY-SA