Английская Википедия:Beta prime distribution

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Шаблон:Short description Шаблон:Probability distribution{B(\alpha,\beta)}\!</math>|

 cdf        =<math> I_{\frac{x}{1+x}(\alpha,\beta) }</math> where <math>I_x(\alpha,\beta)</math> is the incomplete beta function|
 mean       =<math>\frac{\alpha}{\beta-1} \text{ if } \beta>1</math>|
 median     =|
 mode       =<math>\frac{\alpha-1}{\beta+1} \text{ if } \alpha\ge 1\text{, 0 otherwise}\!</math>|
 variance   =<math>\frac{\alpha(\alpha+\beta-1)}{(\beta-2)(\beta-1)^2} \text{ if } \beta>2</math>|
 skewness   =<math>\frac{2(2\alpha+\beta-1)}{\beta-3}\sqrt{\frac{\beta-2}{\alpha(\alpha+\beta-1)}} \text{ if } \beta>3</math>|
 kurtosis   =|
 entropy    =|
 mgf        =Does not exist|
 char       =<math>\frac{e^{-it}\Gamma(\alpha+\beta)}{\Gamma(\beta)}G_{1,2}^{\,2,0}\!\left(\left.{\begin{matrix}\alpha+\beta\\\beta,0\end{matrix}}\;\right|\,-it\right)</math>|

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In probability theory and statistics, the beta prime distribution (also known as inverted beta distribution or beta distribution of the second kind) is an absolutely continuous probability distribution. If <math>p\in[0,1]</math> has a beta distribution, then the odds <math>\frac{p}{1-p}</math> has a beta prime distribution.

Definitions

Beta prime distribution is defined for <math>x > 0</math> with two parameters α and β, having the probability density function:

<math>f(x) = \frac{x^{\alpha-1} (1+x)^{-\alpha -\beta}}{B(\alpha,\beta)}</math>

where B is the Beta function.

The cumulative distribution function is

<math>F(x; \alpha,\beta)=I_{\frac{x}{1+x}}\left(\alpha, \beta \right) ,</math>

where I is the regularized incomplete beta function.

The expected value, variance, and other details of the distribution are given in the sidebox; for <math>\beta>4</math>, the excess kurtosis is

<math>\gamma_2 = 6\frac{\alpha(\alpha+\beta-1)(5\beta-11) + (\beta-1)^2(\beta-2)}{\alpha(\alpha+\beta-1)(\beta-3)(\beta-4) }.</math>

While the related beta distribution is the conjugate prior distribution of the parameter of a Bernoulli distribution expressed as a probability, the beta prime distribution is the conjugate prior distribution of the parameter of a Bernoulli distribution expressed in odds. The distribution is a Pearson type VI distribution.

The mode of a variate X distributed as <math>\beta'(\alpha,\beta)</math> is <math>\hat{X} = \frac{\alpha-1}{\beta+1}</math>. Its mean is <math>\frac{\alpha}{\beta-1}</math> if <math>\beta>1</math> (if <math>\beta \leq 1</math> the mean is infinite, in other words it has no well defined mean) and its variance is <math>\frac{\alpha(\alpha+\beta-1)}{(\beta-2)(\beta-1)^2}</math> if <math>\beta>2</math>.

For <math>-\alpha <k <\beta </math>, the k-th moment <math> E[X^k] </math> is given by

<math> E[X^k]=\frac{B(\alpha+k,\beta-k)}{B(\alpha,\beta)}. </math>

For <math> k\in \mathbb{N} </math> with <math>k <\beta,</math> this simplifies to

<math> E[X^k]=\prod_{i=1}^k \frac{\alpha+i-1}{\beta-i}. </math>

The cdf can also be written as

<math> \frac{x^\alpha \cdot {}_2F_1(\alpha, \alpha+\beta, \alpha+1, -x)}{\alpha \cdot B(\alpha,\beta)}</math>

where <math>{}_2F_1</math> is the Gauss's hypergeometric function ₂F₁ .

Alternative parameterization

The beta prime distribution may also be reparameterized in terms of its mean μ > 0 and precision ν > 0 parameters ( p. 36).

Consider the parameterization μ = α/(β-1) and ν = β- 2, i.e., α = μ( 1 + ν) and β = 2 + ν. Under this parameterization E[Y] = μ and Var[Y] = μ(1 + μ)/ν.

Generalization

Two more parameters can be added to form the generalized beta prime distribution <math>\beta'(\alpha,\beta,p,q)</math>:

• <math>p > 0</math> shape (real)
• <math>q > 0</math> scale (real)

having the probability density function:

<math>f(x;\alpha,\beta,p,q) = \frac{p \left(\frac x q \right)^{\alpha p-1} \left(1+ \left(\frac x q \right)^p\right)^{-\alpha -\beta}}{qB(\alpha,\beta)}</math>

with mean

<math>\frac{q\Gamma\left(\alpha+\tfrac 1 p\right)\Gamma(\beta-\tfrac 1 p)}{\Gamma(\alpha)\Gamma(\beta)} \quad \text{if } \beta p>1</math>

and mode

<math>q \left({\frac{\alpha p -1}{\beta p +1}}\right)^\tfrac{1}{p} \quad \text{if } \alpha p\ge 1</math>

Note that if p = q = 1 then the generalized beta prime distribution reduces to the standard beta prime distribution.

This generalization can be obtained via the following invertible transformation. If <math>y\sim\beta'(\alpha,\beta)</math> and <math>x=qy^{1/p}</math> for <math>q,p>0</math>, then <math>x\sim\beta'(\alpha,\beta,p,q)</math>.

Compound gamma distribution

The compound gamma distribution is the generalization of the beta prime when the scale parameter, q is added, but where p = 1. It is so named because it is formed by compounding two gamma distributions:

<math>\beta'(x;\alpha,\beta,1,q) = \int_0^\infty G(x;\alpha,r)G(r;\beta,q) \; dr</math>

where <math>G(x;a,b)</math> is the gamma pdf with shape <math>a</math> and inverse scale <math>b</math>.

The mode, mean and variance of the compound gamma can be obtained by multiplying the mode and mean in the above infobox by q and the variance by q².

Another way to express the compounding is if <math>r\sim G(\beta,q)</math> and <math>x\mid r\sim G(\alpha,r)</math>, then <math>x\sim\beta'(\alpha,\beta,1,q)</math>. (This gives one way to generate random variates with compound gamma, or beta prime distributions. Another is via the ratio of independent gamma variates, as shown below.)

Properties

• If <math>X \sim \beta'(\alpha,\beta) </math> then <math>\tfrac{1}{X} \sim \beta'(\beta,\alpha)</math>.
• If <math>Y\sim\beta'(\alpha,\beta)</math>, and <math>X=qY^{1/p}</math>, then <math>X\sim\beta'(\alpha,\beta,p,q)</math>.
• If <math>X \sim \beta'(\alpha,\beta,p,q) </math> then <math>kX \sim \beta'(\alpha,\beta,p,kq) </math>.
• <math>\beta'(\alpha,\beta,1,1) = \beta'(\alpha,\beta) </math>
• If <math> X_1 \sim \beta'(\alpha,\beta) </math> and <math> X_2 \sim \beta'(\alpha,\beta) </math> two iid variables, then <math> Y=X_1+X_2 \sim \beta'(\gamma,\delta) </math> with <math> \gamma=\frac{2\alpha(\alpha +\beta^2 - 2\beta + 2\alpha\beta -4\alpha+1)}{(\beta-1)(\alpha+\beta-1)} </math> and <math> \delta =\frac{2\alpha +\beta^2-\beta +2\alpha \beta-4\alpha}{\alpha +\beta -1} </math>, as the beta prime distribution is infinitely divisible.
• More generally, let <math> X_1,...,X_n n</math> iid variables following the same beta prime distribution, i.e. <math>\forall i, 1\leq i\leq n, X_i \sim \beta'(\alpha,\beta)</math>, then the sum <math> S=X_1+...+X_n \sim \beta'(\gamma,\delta) </math> with <math> \gamma=\frac{n\alpha(\alpha +\beta^2 - 2\beta + n\alpha\beta -2n\alpha+1)}{(\beta-1)(\alpha+\beta-1)} </math> and <math> \delta =\frac{2\alpha +\beta^2-\beta +n\alpha \beta-2n\alpha}{\alpha +\beta -1} </math>.

Related distributions

• If <math>X \sim F(2\alpha,2\beta) </math> has an F-distribution, then <math>\tfrac{\alpha}{\beta} X \sim \beta'(\alpha,\beta)</math>, or equivalently, <math>X\sim\beta'(\alpha,\beta , 1 , \tfrac{\beta}{\alpha}) </math>.
• If <math>X \sim \textrm{Beta}(\alpha,\beta)</math> then <math>\frac{X}{1-X} \sim \beta'(\alpha,\beta) </math>.
• If <math>X \sim \beta'(\alpha,\beta)</math> then <math>\frac{X}{1+X} \sim \textrm{Beta}(\alpha,\beta) </math>.
• For gamma distribution parametrization I:
  • If <math>X_k \sim \Gamma(\alpha_k,\theta_k) </math> are independent, then <math>\tfrac{X_1}{X_2} \sim \beta'(\alpha_1,\alpha_2,1,\tfrac{\theta_1}{\theta_2})</math>. Note <math>\theta_1,\theta_2,\tfrac{\theta_1}{\theta_2}</math> are all scale parameters for their respective distributions.
• For gamma distribution parametrization II:
  • If <math>X_k \sim \Gamma(\alpha_k,\beta_k) </math> are independent, then <math>\tfrac{X_1}{X_2} \sim \beta'(\alpha_1,\alpha_2,1,\tfrac{\beta_2}{\beta_1})</math>. The <math>\beta_k</math> are rate parameters, while <math>\tfrac{\beta_2}{\beta_1}</math> is a scale parameter.
  • If <math>\beta_2\sim \Gamma(\alpha_1,\beta_1)</math> and <math>X_2\mid\beta_2\sim\Gamma(\alpha_2,\beta_2)</math>, then <math>X_2\sim\beta'(\alpha_2,\alpha_1,1,\beta_1)</math>. The <math>\beta_k</math> are rate parameters for the gamma distributions, but <math>\beta_1</math> is the scale parameter for the beta prime.
• <math>\beta'(p,1,a,b) = \textrm{Dagum}(p,a,b) </math> the Dagum distribution
• <math>\beta'(1,p,a,b) = \textrm{SinghMaddala}(p,a,b) </math> the Singh–Maddala distribution.
• <math>\beta'(1,1,\gamma,\sigma) = \textrm{LL}(\gamma,\sigma) </math> the log logistic distribution.
• The beta prime distribution is a special case of the type 6 Pearson distribution.
• If X has a Pareto distribution with minimum <math>x_m</math> and shape parameter <math>\alpha</math>, then <math>\dfrac{X}{x_m}-1\sim\beta^\prime(1,\alpha)</math>.
• If X has a Lomax distribution, also known as a Pareto Type II distribution, with shape parameter <math>\alpha</math> and scale parameter <math>\lambda</math>, then <math>\frac{X}{\lambda}\sim \beta^\prime(1,\alpha)</math>.
• If X has a standard Pareto Type IV distribution with shape parameter <math>\alpha</math> and inequality parameter <math>\gamma</math>, then <math>X^{\frac{1}{\gamma}} \sim \beta^\prime(1,\alpha)</math>, or equivalently, <math>X \sim \beta^\prime(1,\alpha,\tfrac{1}{\gamma},1)</math>.
• The inverted Dirichlet distribution is a generalization of the beta prime distribution.
• If <math>X\sim\beta'(\alpha,\beta)</math>, then <math>\ln X</math> has a generalized logistic distribution. More generally, if <math>X\sim\beta'(\alpha,\beta,p,q)</math>, then <math>\ln X</math> has a scaled and shifted generalized logistic distribution.

References

• Johnson, N.L., Kotz, S., Balakrishnan, N. (1995). Continuous Univariate Distributions, Volume 2 (2nd Edition), Wiley. Шаблон:ISBN
• Шаблон:Citation

• MathWorld article