Beta prime distribution#Compound gamma distribution

Beta prime distribution is defined for $x\; >\; 0$ with two parameters α and β, having the probability density function:

: $f(x)\; =\; \backslash frac\{x^\{\backslash alpha-1\}\; (1+x)^\{-\backslash alpha\; -\backslash beta\}\}\{\backslash Beta(\backslash alpha,\backslash beta)\}$

where B is the Beta function.

The cumulative distribution function is

: $F(x;\; \backslash alpha,\backslash beta)=I\_\{\backslash frac\{x\}\{1+x\}\}\backslash left(\backslash alpha,\; \backslash beta\; \backslash right)\; ,$

where I is the regularized incomplete beta function.

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 $\backslash beta\text{'}(\backslash alpha,\backslash beta)$ is $\backslash hat\{X\}\; =\; \backslash frac\{\backslash alpha-1\}\{\backslash beta+1\}$.

Its mean is $\backslash frac\{\backslash alpha\}\{\backslash beta-1\}$ if $\backslash beta>1$ (if $\backslash beta\; \backslash leq\; 1$ the mean is infinite, in other words it has no well defined mean) and its variance is $\backslash frac\{\backslash alpha(\backslash alpha+\backslash beta-1)\}\{(\backslash beta-2)(\backslash beta-1)^2\}$ if $\backslash beta>2$.

For k-th moment $E[X^k]$ is given by

:$E[X^k]=\backslash frac\{\backslash Beta(\backslash alpha+k,\backslash beta-k)\}\{\backslash Beta(\backslash alpha,\backslash beta)\}.$

For $k\backslash in\; \backslash mathbb\{N\}$ with this simplifies to

:$E[X^k]=\backslash prod\_\{i=1\}^k\; \backslash frac\{\backslash alpha+i-1\}\{\backslash beta-i\}.$

The cdf can also be written as

:$\backslash frac\{x^\backslash alpha\; \backslash cdot\; \{\}\_2F\_1(\backslash alpha,\; \backslash alpha+\backslash beta,\; \backslash alpha+1,\; -x)\}\{\backslash alpha\; \backslash cdot\; \backslash Beta(\backslash alpha,\backslash beta)\}$

where $\{\}\_2F\_1$ is the Gauss's hypergeometric function ₂F₁ .

The beta prime distribution may also be reparameterized in terms of its mean μ > 0 and precision ν > 0 parameters ({{cite journal |last1=Bourguignon |first1=M. |author2=Santos-Neto, M. |author3=de Castro, M. |year=2021 |title=A new regression model for positive random variables with skewed and long tail |journal=Metron |volume=79| pages=33–55 |doi=10.1007/s40300-021-00203-y|s2cid=233534544 }} 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.

Two more parameters can be added to form the generalized beta prime distribution $\backslash beta\text{'}(\backslash alpha,\backslash beta,p,q)$:

• $p\; >\; 0$ shape (real)
• $q\; >\; 0$ scale (real)

having the probability density function:

: $f(x;\backslash alpha,\backslash beta,p,q)\; =\; \backslash frac\{p\; \backslash left(\backslash frac\; x\; q\; \backslash right)^\{\backslash alpha\; p-1\}\; \backslash left(1+\; \backslash left(\backslash frac\; x\; q\; \backslash right)^p\backslash right)^\{-\backslash alpha\; -\backslash beta\}\}\{q\; \backslash Beta(\backslash alpha,\backslash beta)\}$

with mean

: $\backslash frac\{q\backslash Gamma\backslash left(\backslash alpha+\backslash tfrac\; 1\; p\backslash right)\backslash Gamma(\backslash beta-\backslash tfrac\; 1\; p)\}\{\backslash Gamma(\backslash alpha)\backslash Gamma(\backslash beta)\}\; \backslash quad\; \backslash text\{if\; \}\; \backslash beta\; p>1$

and mode

: $q\; \backslash left(\{\backslash frac\{\backslash alpha\; p\; -1\}\{\backslash beta\; p\; +1\}\}\backslash right)^\backslash tfrac\{1\}\{p\}\; \backslash quad\; \backslash text\{if\; \}\; \backslash alpha\; p\backslash ge\; 1$

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 $y\backslash sim\backslash beta\text{'}(\backslash alpha,\backslash beta)$ and $x=qy^\{1/p\}$ for $q,p>0$, then $x\backslash sim\backslash beta\text{'}(\backslash alpha,\backslash beta,p,q)$.

The compound gamma distribution{{cite journal|last=Dubey|first=Satya D.|title=Compound gamma, beta and F distributions|journal=Metrika|date=December 1970|volume=16|pages=27–31|doi=10.1007/BF02613934|s2cid=123366328}} 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:

:$\backslash beta\text{'}(x;\backslash alpha,\backslash beta,1,q)\; =\; \backslash int\_0^\backslash infty\; G(x;\backslash alpha,r)G(r;\backslash beta,q)\; \backslash ;\; dr$

where $G(x;a,b)$ is the gamma pdf with shape $a$ and inverse scale $b$.

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 $r\backslash sim\; G(\backslash beta,q)$ and $x\backslash mid\; r\backslash sim\; G(\backslash alpha,r)$, then $x\backslash sim\backslash beta\text{'}(\backslash alpha,\backslash beta,1,q)$. 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.

• If $X\; \backslash sim\; \backslash beta\text{'}(\backslash alpha,\backslash beta)$ then $\backslash tfrac\{1\}\{X\}\; \backslash sim\; \backslash beta\text{'}(\backslash beta,\backslash alpha)$.
• If $Y\backslash sim\backslash beta\text{'}(\backslash alpha,\backslash beta)$, and $X=qY^\{1/p\}$, then $X\backslash sim\backslash beta\text{'}(\backslash alpha,\backslash beta,p,q)$.
• If $X\; \backslash sim\; \backslash beta\text{'}(\backslash alpha,\backslash beta,p,q)$ then $kX\; \backslash sim\; \backslash beta\text{'}(\backslash alpha,\backslash beta,p,kq)$.
• $\backslash beta\text{'}(\backslash alpha,\backslash beta,1,1)\; =\; \backslash beta\text{'}(\backslash alpha,\backslash beta)$

• If $X\; \backslash sim\; \backslash textrm\{Beta\}(\backslash alpha,\backslash beta)$, then $\backslash frac\{X\}\{1-X\}\; \backslash sim\; \backslash beta\text{'}(\backslash alpha,\backslash beta)$. This property can be used to generate beta prime distributed variates.
• If $X\; \backslash sim\; \backslash beta\text{'}(\backslash alpha,\backslash beta)$, then $\backslash frac\{X\}\{1+X\}\; \backslash sim\; \backslash textrm\{Beta\}(\backslash alpha,\backslash beta)$. This is a corollary from the property above.
• If $X\; \backslash sim\; F(2\backslash alpha,2\backslash beta)$ has an F-distribution, then $\backslash tfrac\{\backslash alpha\}\{\backslash beta\}\; X\; \backslash sim\; \backslash beta\text{'}(\backslash alpha,\backslash beta)$, or equivalently, $X\backslash sim\backslash beta\text{'}(\backslash alpha,\backslash beta\; ,\; 1\; ,\; \backslash tfrac\{\backslash beta\}\{\backslash alpha\})$.
• For gamma distribution parametrization I:
• If $X\_k\; \backslash sim\; \backslash Gamma(\backslash alpha\_k,\backslash theta\_k)$ are independent, then $\backslash tfrac\{X\_1\}\{X\_2\}\; \backslash sim\; \backslash beta\text{'}(\backslash alpha\_1,\backslash alpha\_2,1,\backslash tfrac\{\backslash theta\_1\}\{\backslash theta\_2\})$. Note $\backslash theta\_1,\backslash theta\_2,\backslash tfrac\{\backslash theta\_1\}\{\backslash theta\_2\}$ are all scale parameters for their respective distributions.
• For gamma distribution parametrization II:
• If $X\_k\; \backslash sim\; \backslash Gamma(\backslash alpha\_k,\backslash beta\_k)$ are independent, then $\backslash tfrac\{X\_1\}\{X\_2\}\; \backslash sim\; \backslash beta\text{'}(\backslash alpha\_1,\backslash alpha\_2,1,\backslash tfrac\{\backslash beta\_2\}\{\backslash beta\_1\})$. The $\backslash beta\_k$ are rate parameters, while $\backslash tfrac\{\backslash beta\_2\}\{\backslash beta\_1\}$ is a scale parameter.
• If $\backslash beta\_2\backslash sim\; \backslash Gamma(\backslash alpha\_1,\backslash beta\_1)$ and $X\_2\backslash mid\backslash beta\_2\backslash sim\backslash Gamma(\backslash alpha\_2,\backslash beta\_2)$, then $X\_2\backslash sim\backslash beta\text{'}(\backslash alpha\_2,\backslash alpha\_1,1,\backslash beta\_1)$. The $\backslash beta\_k$ are rate parameters for the gamma distributions, but $\backslash beta\_1$ is the scale parameter for the beta prime.
• $\backslash beta\text{'}(p,1,a,b)\; =\; \backslash textrm\{Dagum\}(p,a,b)$ the Dagum distribution
• $\backslash beta\text{'}(1,p,a,b)\; =\; \backslash textrm\{SinghMaddala\}(p,a,b)$ the Singh–Maddala distribution.
• $\backslash beta\text{'}(1,1,\backslash gamma,\backslash sigma)\; =\; \backslash textrm\{LL\}(\backslash gamma,\backslash sigma)$ 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 $x\_m$ and shape parameter $\backslash alpha$, then $\backslash dfrac\{X\}\{x\_m\}-1\backslash sim\backslash beta^\backslash prime(1,\backslash alpha)$.
• If X has a Lomax distribution, also known as a Pareto Type II distribution, with shape parameter $\backslash alpha$ and scale parameter $\backslash lambda$, then $\backslash frac\{X\}\{\backslash lambda\}\backslash sim\; \backslash beta^\backslash prime(1,\backslash alpha)$.
• If X has a standard Pareto Type IV distribution with shape parameter $\backslash alpha$ and inequality parameter $\backslash gamma$, then $X^\{\backslash frac\{1\}\{\backslash gamma\}\}\; \backslash sim\; \backslash beta^\backslash prime(1,\backslash alpha)$, or equivalently, $X\; \backslash sim\; \backslash beta^\backslash prime(1,\backslash alpha,\backslash tfrac\{1\}\{\backslash gamma\},1)$.
• The inverted Dirichlet distribution is a generalization of the beta prime distribution.
• If $X\backslash sim\backslash beta\text{'}(\backslash alpha,\backslash beta)$, then $\backslash ln\; X$ has a generalized logistic distribution. More generally, if $X\backslash sim\backslash beta\text{'}(\backslash alpha,\backslash beta,p,q)$, then $\backslash ln\; X$ has a scaled and shifted generalized logistic distribution.
• If $X\backslash sim\backslash beta\text{'}\backslash left(\backslash frac\{1\}\{2\},\backslash frac\{1\}\{2\}\backslash right)$, then $\backslash pm\backslash sqrt\{X\}$ follows a Cauchy distribution, which is equivalent to a student-t distribution with the degrees of freedom of 1.

{{Reflist}}

• Johnson, N.L., Kotz, S., Balakrishnan, N. (1995). Continuous Univariate Distributions, Volume 2 (2nd Edition), Wiley. {{ISBN|0-471-58494-0}}
• {{Citation|last1=Bourguignon| first1=M.| last2=Santos-Neto| first2=M.| last3=de Castro| first3=M.| year=2021| title= A new regression model for positive random variables with skewed and long tail| journal=Metron|volume=79| pages=33–55|doi=10.1007/s40300-021-00203-y | s2cid=233534544}}

• [http://mathworld.wolfram.com/BetaPrimeDistribution.html MathWorld article]

{{ProbDistributions|continuous-semi-infinite}}

Category:Continuous distributions

Category:Compound probability distributions