beta negative binomial distribution

{{Probability distribution

| name = Beta Negative Binomial

| type = mass

| pdf_image =

| cdf_image =

| notation =

| parameters = $\alpha > 0$ shape (real)
$\beta > 0$ shape (real)
$r > 0$ — number of successes until the experiment is stopped (integer but can be extended to real)

| support = $k \in \{0,1,2,\ldots\}$

| pdf = $\frac{\Beta(r+k,\alpha+\beta)}{\Beta(r,\alpha)}\frac{\Gamma(k+\beta)}{k!\;\Gamma(\beta)}$

| cdf =

| mean = $\begin{cases}
\frac{r\beta}{\alpha-1} & \text{if}\ \alpha>1 \\
\infty & \text{otherwise}\ \end{cases}$

| median =

| mode =

| variance = $\begin{cases}
\frac{r\beta(r+\alpha-1)(\beta+\alpha-1)}{(\alpha-2){(\alpha-1)}^2} & \text{if}\ \alpha>2 \\
\infty & \text{otherwise}\ \end{cases}$

| skewness = $\begin{cases}
\frac{(2r+\alpha-1)(2\beta+\alpha-1)}{(\alpha-3)\sqrt{\frac{r\beta(r+\alpha-1)(\beta+\alpha-1)}{\alpha-2}}} & \text{if}\ \alpha>3 \\
\infty & \text{otherwise}\ \end{cases}$

| kurtosis =

| entropy =

| mgf = does not exist

| char = ${}_{2}F_{1}(\beta,r;\alpha+\beta+r;e^{it}) \frac{(\alpha)^{(r)}}{(\alpha+\beta)^{(r)}} \!$ where $(x)^{(r)} = \frac{\Gamma(x+r)}{\Gamma(x)}$ is the Pochhammer symbol and ${}_{2}F_{1}$ is the hypergeometric function.

| pgf = ${}_{2}F_{1}(\beta,r;\alpha+\beta+r;z) \frac{(\alpha)^{(r)}}{(\alpha+\beta)^{(r)}}$

}}

In probability theory, a beta negative binomial distribution is the probability distribution of a discrete random variable $X$ equal to the number of failures needed to get $r$ successes in a sequence of independent Bernoulli trials. The probability $p$ of success on each trial stays constant within any given experiment but varies across different experiments following a 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 distributionJohnson et al. (1993) or simply abbreviated as the BNB distribution. A shifted form of the distribution has been called the beta-Pascal distribution.

If parameters of the beta distribution are $\alpha$ and $\beta$ , and if

: $X \mid p \sim \mathrm{NB}(r,p),$

where

: $p \sim \textrm{B}(\alpha,\beta),$

then the marginal distribution of $X$ (i.e. the posterior predictive distribution) is a beta negative binomial distribution:

: $X \sim \mathrm{BNB}(r,\alpha,\beta).$

In the above, $\mathrm{NB}(r,p)$ is the negative binomial distribution and $\textrm{B}(\alpha,\beta)$ is the beta distribution.

Definition and derivation

Denoting $f_{X|p}(k|q), f_{p}(q|\alpha,\beta)$ the densities of the negative binomial and beta distributions respectively, we obtain the PMF $f(k|\alpha,\beta,r)$ of the BNB distribution by marginalization:

: $\begin{align}
f(k|\alpha,\beta,r) \; =& \; \int_0^1 f_{X|p}(k|r,q) \cdot f_{p}(q|\alpha,\beta) \mathrm{d} q \\
=& \; \int_0^1 \binom{k+r-1}{k} (1-q)^k q^r \cdot \frac{q^{\alpha-1}(1-q)^{\beta-1}} {\Beta(\alpha,\beta)} \mathrm{d} q \\
=& \; \frac{1}{\Beta(\alpha,\beta)} \binom{k+r-1}{k} \int_0^1 q^{\alpha+r-1}(1-q)^{\beta+k-1} \mathrm{d} q
\end{align}$

Noting that the integral evaluates to:

: $\int_0^1 q^{\alpha+r-1}(1-q)^{\beta+k-1} \mathrm{d} q = \frac{\Gamma(\alpha+r)\Gamma(\beta+k)}{\Gamma(\alpha+\beta+k+r)}$

we can arrive at the following formulas by relatively simple manipulations.

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

: $f(k|\alpha,\beta,r)=\binom{r+k-1}k\frac{\Beta(\alpha+r,\beta+k)}{\Beta(\alpha,\beta)}$ .

More generally, the PMF can be written

: $f(k|\alpha,\beta,r)=\frac{\Gamma(r+k)}{k!\;\Gamma(r)}\frac{\Beta(\alpha+r,\beta+k)}{\Beta(\alpha,\beta)}$

: $f(k|\alpha,\beta,r)=\frac{\Beta(r+k,\alpha+\beta)}{\Beta(r,\alpha)}\frac{\Gamma(k+\beta)}{k!\;\Gamma(\beta)}$ .

=PMF expressed with Gamma=

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

: $f(k|\alpha,\beta,r)=\binom{r+k-1}k\frac{\Gamma(\alpha+r)\Gamma(\beta+k)\Gamma(\alpha+\beta)}{\Gamma(\alpha+r+\beta+k)\Gamma(\alpha)\Gamma(\beta)}$ .

More generally, the PMF can be written as

: $f(k|\alpha,\beta,r)=\frac{\Gamma(r+k)}{k!\;\Gamma(r)}\frac{\Gamma(\alpha+r)\Gamma(\beta+k)\Gamma(\alpha+\beta)}{\Gamma(\alpha+r+\beta+k)\Gamma(\alpha)\Gamma(\beta)}$ .

=PMF expressed with the rising Pochammer symbol=

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

: $f(k|\alpha,\beta,r)=\frac{r^{(k)}\alpha^{(r)}\beta^{(k)}}{k!(\alpha+\beta)^{(r+k)}}$

Properties

=Factorial Moments=

The {{math|k}}-th factorial moment of a beta negative binomial random variable {{math|X}} is defined for $k < \alpha$ and in this case is equal to

: $\operatorname{E}\bigl[(X)_k\bigr] = \frac{\Gamma(r+k)}{\Gamma(r)}\frac{\Gamma(\beta+k)}{\Gamma(\beta)}\frac{\Gamma(\alpha-k)}{\Gamma(\alpha)}.$

=Non-identifiable=

The beta negative binomial is non-identifiable which can be seen easily by simply swapping $r$ and $\beta$ in the above density or characteristic function and noting that it is unchanged. Thus estimation demands that a constraint be placed on $r$ , $\beta$ or both.

=Relation to other distributions=

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

=Heavy tailed=

By Stirling's approximation to the beta function, it can be easily shown that for large $k$

: $f(k|\alpha,\beta,r) \sim \frac{\Gamma(\alpha+r)}{\Gamma(r)\Beta(\alpha,\beta)}\frac{k^{r-1}}{(\beta+k)^{r+\alpha}}$

which implies that the beta negative binomial distribution is heavy tailed and that moments less than or equal to $\alpha$ do not exist.

Beta geometric distribution

The beta geometric distribution is an important special case of the beta negative binomial distribution occurring for $r=1$ . In this case the pmf simplifies to

: $f(k|\alpha,\beta)=\frac{\mathrm{B}(\alpha+1,\beta+k)} {\mathrm{B}(\alpha,\beta)}$ .

This distribution is used in some Buy Till you Die (BTYD) models.

Further, when $\beta=1$ the beta geometric reduces to the 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 $X \sim BG(\alpha,1)$ then $X+1 \sim YS(\alpha)$ .

Beta negative binomial as a Pólya urn model

In the case when the 3 parameters $r, \alpha$ and $\beta$ are positive integers, the Beta negative binomial can also be motivated by an urn model - or more specifically a basic Pólya urn model. Consider an urn initially containing $\alpha$ red balls (the stopping color) and $\beta$ blue balls. At each step of the model, a ball is drawn at random from the urn and replaced, along with one additional ball of the same color. The process is repeated over and over, until $r$ red colored balls are drawn. The random variable $X$ of observed draws of blue balls are distributed according to a $\mathrm{BNB}(r, \alpha, \beta)$ . Note, at the end of the experiment, the urn always contains the fixed number $r+\alpha$ of red balls while containing the random number $X+\beta$ blue balls.

By the non-identifiability property, $X$ can be equivalently generated with the urn initially containing $\alpha$ red balls (the stopping color) and $r$ blue balls and stopping when $\beta$ red balls are observed.

Notes

References

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}}

External links

Interactive graphic: [http://www.math.wm.edu/~leemis/chart/UDR/UDR.html Univariate Distribution Relationships]

Category:Discrete distributions

Category:Compound probability distributions

Category:Factorial and binomial topics