Английская Википедия:Beta negative binomial distribution
Шаблон:Probability distribution} & \text{if}\ \alpha>3 \\
\infty & \text{otherwise}\ \end{cases}</math>
| kurtosis =
| entropy =
| mgf = does not exist
| char = <math> \frac{\Gamma(\alpha+r)\Gamma(\alpha+\beta)} {\Gamma(\alpha+\beta+r)\Gamma(\alpha)} {}_{2}F_{1}(r,\beta;\alpha+\beta+r;e^{it})\!</math> where <math>\Gamma</math> is the gamma function and <math>{}_{2}F_{1}</math> is the hypergeometric function.}}
In probability theory, a beta negative binomial distribution is the probability distribution of a discrete random variable <math>X</math> equal to the number of failures needed to get <math>r</math> successes in a sequence of independent Bernoulli trials. The probability <math>p</math> 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 distribution[1] or simply abbreviated as the BNB distribution. A shifted form of the distribution has been called the beta-Pascal distribution.[1]
If parameters of the beta distribution are <math>\alpha</math> and <math>\beta</math>, and if
- <math>
X \mid p \sim \mathrm{NB}(r,p), </math> where
- <math>
p \sim \textrm{B}(\alpha,\beta),
</math> then the marginal distribution of <math>X</math> is a beta negative binomial distribution:
- <math>
X \sim \mathrm{BNB}(r,\alpha,\beta). </math>
In the above, <math>\mathrm{NB}(r,p)</math> is the negative binomial distribution and <math>\textrm{B}(\alpha,\beta)</math> is the beta distribution.
Definition and derivation
Denoting <math>f_{X|p}(k|q), f_{p}(q|\alpha,\beta)</math> the densities of the negative binomial and beta distributions respectively, we obtain the PMF <math>f(k|\alpha,\beta,r)</math> of the BNB distribution by marginalization:
- <math>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</math>
Noting that the integral evaluates to:
- <math> \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)}</math>
we can arrive at the following formulas by relatively simple manipulations.
If <math>r</math> is an integer, then the PMF can be written in terms of the beta function,:
- <math>f(k|\alpha,\beta,r)=\binom{r+k-1}k\frac{\Beta(\alpha+r,\beta+k)}{\Beta(\alpha,\beta)}</math>.
More generally, the PMF can be written
- <math>f(k|\alpha,\beta,r)=\frac{\Gamma(r+k)}{k!\;\Gamma(r)}\frac{\Beta(\alpha+r,\beta+k)}{\Beta(\alpha,\beta)}</math>
or
- <math>f(k|\alpha,\beta,r)=\frac{\Beta(r+k,\alpha+\beta)}{\Beta(r,\alpha)}\frac{\Gamma(k+\beta)}{k!\;\Gamma(\beta)}</math>.
PMF expressed with Gamma
Using the properties of the Beta function, the PMF with integer <math>r</math> can be rewritten as:
- <math>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)}</math>.
More generally, the PMF can be written as
- <math>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)}</math>.
PMF expressed with the rising Pochammer symbol
The PMF is often also presented in terms of the Pochammer symbol for integer <math>r</math>
- <math>f(k|\alpha,\beta,r)=\frac{r^{(k)}\alpha^{(r)}\beta^{(k)}}{k!(\alpha+\beta)^{(r+k)}}</math>
Properties
Factorial Moments
The Шаблон:Math-th factorial moment of a beta negative binomial random variable Шаблон:Math is defined for <math>k < \alpha</math> and in this case is equal to
- <math>\operatorname{E}\bigl[(X)_k\bigr] = \frac{\Gamma(r+k)}{\Gamma(r)}\frac{\Gamma(\beta+k)}{\Gamma(\beta)}\frac{\Gamma(\alpha-k)}{\Gamma(\alpha)}.
</math>
Non-identifiable
The beta negative binomial is non-identifiable which can be seen easily by simply swapping <math>r</math> and <math>\beta</math> in the above density or characteristic function and noting that it is unchanged. Thus estimation demands that a constraint be placed on <math>r</math>, <math>\beta</math> or both.
Relation to other distributions
The beta negative binomial distribution contains the beta geometric distribution as a special case when either <math>r=1</math> or <math>\beta=1</math>. It can therefore approximate the geometric distribution arbitrarily well. It also approximates the negative binomial distribution arbitrary well for large <math>\alpha</math>. It can therefore approximate the Poisson distribution arbitrarily well for large <math>\alpha</math>, <math>\beta</math> and <math>r</math>.
Heavy tailed
By Stirling's approximation to the beta function, it can be easily shown that for large <math>k</math>
- <math>f(k|\alpha,\beta,r) \sim \frac{\Gamma(\alpha+r)}{\Gamma(r)\Beta(\alpha,\beta)}\frac{k^{r-1}}{(\beta+k)^{r+\alpha}}</math>
which implies that the beta negative binomial distribution is heavy tailed and that moments less than or equal to <math>\alpha</math> do not exist.
Beta geometric distribution
The beta geometric distribution is an important special case of the beta negative binomial distribution occurring for <math>r=1 </math>. In this case the pmf simplifies to
- <math>f(k|\alpha,\beta)=\frac{\mathrm{B}(\alpha+1,\beta+k)} {\mathrm{B}(\alpha,\beta)}</math>.
This distribution is used in some Buy Till you Die (BTYD) models.
Further, when <math> \beta=1</math> 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 <math> X \sim BG(\alpha,1) </math> then <math> X+1 \sim YS(\alpha)</math>.
Beta negative binomial as a Pólya urn model
In the case when the 3 parameters <math>r, \alpha</math> and <math>\beta</math> 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 <math>\alpha</math> red balls (the stopping color) and <math>\beta</math> 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 <math>r</math> red colored balls are drawn. The random variable <math>X</math> of observed draws of blue balls are distributed according to a <math>\mathrm{BNB}(r, \alpha, \beta)</math>. Note, at the end of the experiment, the urn always contains the fixed number <math>r+\alpha</math> of red balls while containing the random number <math>X+\beta</math> blue balls.
By the non-identifiability property, <math>X</math> can be equivalently generated with the urn initially containing <math>\alpha</math> red balls (the stopping color) and <math>r</math> blue balls and stopping when <math>\beta</math> red balls are observed.
See also
Notes
References
- Johnson, N.L.; Kotz, S.; Kemp, A.W. (1993) Univariate Discrete Distributions, 2nd edition, Wiley Шаблон:ISBN (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 Шаблон:Doi
External links
- Interactive graphic: Univariate Distribution Relationships
