Beta-negative binomial distribution (Waring)

class ctx.dist_beta_negbinomial(r, alpha, beta)

where ctx is fpm, mpm, ipm, dec, gmp or apm.

The 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, while constant within any given experiment, is itself a random variable following a beta distribution, varying between different experiments. Thus the distribution is a compound probability distribution.

See also: Wikipedia [1313], Johnson et al. [411] page 256.

The beta-negative binomial distribution (like the beta-binomial distribution) deals with draws where the probability \(p\) of success on each trial, while constant within any given experiment, is itself a random variable following a beta distribution, varying between different experiments, so that the probability of success is different in each draw. In contrast, the negative-binomial distribution (like the binomial distribution) deals with draws where the probability of success is the same and the trials are independent. The following table summarizes the four distributions related to drawing items and \(p\):

Category	\(p\) is a constant	\(p\) is a beta variable
# of successes in constant # of draws	binomial distribution	betabinomial distribution
# of successes in constant # of failures	negative binomial distribution	beta-negative binomial distribution

This distribution has also been called both the inverse Markov-Pólya distribution and the generalized Waring distribution.

dist_beta_negbinomial.pmf(x)

Returns \(\text{pmf}_X(x)\), the probability mass function (pmf) of a random variable \(X\), following a beta-negative binomial distribution:

\[\text{pmf}_X(x) = \binom{-\beta}{x} \binom{\alpha+\beta-1}{-r-x} \bigg/ \binom{\alpha-1}{-r}.\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print ("pmf: ", hypergeometric(mu, sigma).pmf(x))
6.3563523462564525615615615614561356E-20

dist_beta_negbinomial.cdf(x)

Returns \(\text{cdf}_X(x)\), the cumulative distribution function (cdf) of a random variable \(X\), following a beta-negative binomial distribution:

\[\text{cdf}_X(k) = \sum_{j=\max(0,n+K-N)}^{k} \text{pmf}_X(j) = 1 - \text{pmf}_X(k+1) \times {}_3F_2(1,k+1-K,k+1-n;k+2,N+k+2-K-n;1),\]

where \({}_3F_2(\cdot)\) is a generalized hypergeometric function (see hyp3f2().)

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print ("cdf: ", hypergeometric(mu, sigma).pmf(x))
6.3563523462564525615615615614561356E-20

dist_beta_negbinomial.sf(x)

Returns \(\text{sf}_X(x)\), the survival function (sf) of a random variable \(X\), following a beta-negative binomial distribution:

\[\text{sf}_X(k) = \sum_{j=k+1}^{\min(K,n)} \text{pmf}_X(j) = \text{pmf}_X(k+1) \times {}_3F_2(1,k+1-K,k+1-n;k+2,N+k+2-K-n;1),\]

where \({}_3F_2(\cdot)\) is a generalized hypergeometric function (see hyp3f2().)

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print (" sf: ", hypergeometric(mu, sigma).pmf(x))
sf: 6.3563523462564525615615615614561356E-20

dist_beta_negbinomial.qtf(q)

Returns \(\text{qtf}_X(x)\), the quantile function (qtf) of a random variable \(X\), following a beta-negative binomial distribution.

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; q = 0.3;
>>> print ("qtf: ", hypergeometric(mu, sigma).qtf(q))
qtf: 6.3563523462564525615615615614561356E+00

dist_beta_negbinomial.isf(q)

Returns \(\text{isf}_X(q)\), the inverse survival function (isf) of a random variable \(X\), following a beta-negative binomial distribution:

\[\text{isf}_X(q) = \text{qtf}_X(1-q).\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; q = 0.3;
>>> print ("isf: ", hypergeometric(mu, sigma).isf(q))
6.3563523462564525615615615614561356E+00

dist_beta_negbinomial.g_x(t)

Returns \(G_X(t)\), the probability generating function of a random variable \(X\), following a beta-negative binomial distribution:

\[G_X(t) = \frac{{}_2F_1(r, \beta; \alpha+\beta+r; t)}{{}_2F_1(r, \beta; \alpha+\beta+r; 1)}\]

where \({}_2F_1(\cdot)\) is the Gauss hypergeometric function (see hyp2f1().)

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; t = 0.3;
>>> print ("c_x: ", hypergeometric(mu, sigma).c_x(t))
6.3563523462564525615615615614561356E+00

dist_beta_negbinomial.c_x(t)

Returns \(C_X(t)\), the characteristic function of a random variable \(X\), following a beta-negative binomial distribution:

\[C_X(t) = \frac{{}_2F_1(r, \beta; \alpha+\beta+r; e^{it})}{{}_2F_1(r, \beta; \alpha+\beta+r; 1)}\]

where \({}_2F_1(\cdot)\) is the Gauss hypergeometric function (see hyp2f1().)

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; t = 0.3;
>>> print ("c_x: ", hypergeometric(mu, sigma).c_x(t))
6.3563523462564525615615615614561356E+00

dist_beta_negbinomial.m_x(t)

Returns \(M_X(t)\), the moment generating function of a random variable \(X\), following a beta-negative binomial distribution:

\[M_X(t) = \frac{{}_2F_1(r, \beta; \alpha+\beta+r; e^{t})}{{}_2F_1(r, \beta; \alpha+\beta+r; 1)}\]

where \({}_2F_1(\cdot)\) is the Gauss hypergeometric function (see hyp2f1().)

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; t = 0.3;
>>> print ("m_x: ", hypergeometric(mu, sigma).c_x(t))
6.3563523462564525615615615614561356E+00

dist_beta_negbinomial.k_x(t, k=0)

Returns \(K_X(t)\), the cumulant generating function of a random variable \(X\), following a beta-negative binomial distribution:

\[K_X(t) = \log \left[ \frac{{}_2F_1(r, \beta; \alpha+\beta+r; e^{t})}{{}_2F_1(r, \beta; \alpha+\beta+r; 1)} \right].\]

where \({}_2F_1(\cdot)\) is the Gauss hypergeometric function (see hyp2f1().)

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; t = 0.3; k = 6;
>>> print ("c_x: ", hypergeometric(mu, sigma).k_x(t, k))
6.3563523462564525615615615614561356E+00

dist_beta_negbinomial.moments(k)

Returns the first \(j\) raw moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following a beta-negative binomial distribution. The raw moments are calculated from the factorial moments:

\[\mu'_{[r]} = \frac{n! a! (a+b-r)! }{(n-r)! (a-r)! (a+b)!}\]

\[\mu'_{[r]} = \frac{n!}{(n-r)!} \frac{a!}{(a-r)!} \frac{(a+b-r)!}{(a+b)!}.\]

For \(n>0\) and \(a>0\), when \(n \le r\) or \(a \le r\) then \(\mu'_{[r]} = 0\).

When \(a<0\) and \(b<0\) with \(b\) an integer

\[\frac{a!}{(a+b)!} = \frac{(-1)^b (-a-b-1)!}{(-a-1)!}\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; k = 6;
>>> print ("saddlepoint: ", hypergeometric(mu, sigma).moments(k))
6.3563523462564525615615615614561356E+00

dist_beta_negbinomial.cumulants(k)

Returns the first \(j\) cumulants, \(\kappa_j, j = 1 \ldots k\), of a random variable \(X\), following a beta-negative binomial distribution. The cumulants are calculated from the moments.

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; k = 6;
>>> print ("saddlepoint: ", hypergeometric(mu, sigma).cumulants(k))
6.3563523462564525615615615614561356E+00