Beta prime distribution

Beta prime
Beta prime
	Probability density function
	Cumulative distribution function
Parameters	α>0 shape (real); β>0 shape (real)
Support	x∈[0,∞)
PDF	f(x)=xα−1(1+x)−α−βB(α,β)
CDF	Ix1+x(α,β) where Ix(α,β) is the regularized incomplete beta function
Mean	αβ−1 if β>1
Mode	α−1β+1 if α≥1, 0 otherwise
Variance	α(α+β−1)(β−2)(β−1)2 if β>2
Skewness	2(2α+β−1)β−3β−2α(α+β−1) if β>3
Excess kurtosis	6α(α+β−1)(5β−11)+(β−1)2(β−2)α(α+β−1)(β−3)(β−4) if β>4
Entropy	log⁡(B(α,β))+(α−1)(ψ(β)−ψ(α))+(α+β)(ψ(1−α−β)−ψ(1−β)+πsin⁡(απ)sin⁡(βπ)sin⁡((α+β)π))) where ψ is the digamma function.
MGF	Does not exist
CF	e−itΓ(α+β)Γ(β)G1,22,0(α+ββ,0\|−it)

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

Definitions

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

f (x) = \frac{x^{α - 1} (1 + x)^{- α - β}}{B (α, β)}

where B is the Beta function.

The cumulative distribution function is

F (x; α, β) = I_{\frac{x}{1 + x}} (α, β),

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.^[1]

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

For $- α < k < β$ , the k-th moment $E [X^{k}]$ is given by

E [X^{k}] = \frac{B (α + k, β - k)}{B (α, β)} .

For $k \in ℕ$ with $k < β,$ this simplifies to

E [X^{k}] = \prod_{i = 1}^{k} \frac{α + i - 1}{β - i} .

The cdf can also be written as

\frac{x^{α} \cdot_{2} F_{1} (α, α + β, α + 1, - x)}{α \cdot B (α, β)}

where $_{2} F_{1}$ 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 (^[2] 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 $β^{'} (α, β, p, q)$ :

$p > 0$ shape (real)
$q > 0$ scale (real)

having the probability density function:

f (x; α, β, p, q) = \frac{p {(\frac{x}{q})}^{α p - 1} {(1 + {(\frac{x}{q})}^{p})}^{- α - β}}{q B (α, β)}

with mean

\frac{q Γ (α + \frac{1}{p}) Γ (β - \frac{1}{p})}{Γ (α) Γ (β)} if β p > 1

and mode

q {(\frac{α p - 1}{β p + 1})}^{\frac{1}{p}} if α p \geq 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 $y \sim β^{'} (α, β)$ and $x = q y^{1 / p}$ for $q, p > 0$ , then $x \sim β^{'} (α, β, p, q)$ .

Compound gamma distribution

The compound gamma distribution^[3] 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:

β^{'} (x; α, β, 1, q) = \int_{0}^{\infty} G (x; α, r) G (r; β, q) d r

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 $r \sim G (β, q)$ and $x ∣ r \sim G (α, r)$ , then $x \sim β^{'} (α, β, 1, q)$ . 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.

Properties

If $X \sim β^{'} (α, β)$ then $\frac{1}{X} \sim β^{'} (β, α)$ .
If $Y \sim β^{'} (α, β)$ , and $X = q Y^{1 / p}$ , then $X \sim β^{'} (α, β, p, q)$ .
If $X \sim β^{'} (α, β, p, q)$ then $k X \sim β^{'} (α, β, p, k q)$ .
$β^{'} (α, β, 1, 1) = β^{'} (α, β)$

Related distributions

If $X \sim Beta (α, β)$ , then $\frac{X}{1 - X} \sim β^{'} (α, β)$ and $\frac{1}{X} - 1 \sim β^{'} (β, α)$ . This property can be used to generate beta prime distributed variates.
If $X \sim β^{'} (α, β)$ , then $\frac{X}{1 + X} \sim Beta (α, β)$ and $\frac{1}{X + 1} \sim Beta (β, α)$ . This is a corollary from the property above.
If $X \sim F (2 α, 2 β)$ has an F-distribution, then $\frac{α}{β} X \sim β^{'} (α, β)$ , or equivalently, $X \sim β^{'} (α, β, 1, \frac{β}{α})$ .
For gamma distribution parametrization I:
- If $X_{k} \sim Γ (α_{k}, θ_{k})$ are independent, then $\frac{X_{1}}{X_{2}} \sim β^{'} (α_{1}, α_{2}, 1, \frac{θ_{1}}{θ_{2}})$ . Note $θ_{1}, θ_{2}, \frac{θ_{1}}{θ_{2}}$ are all scale parameters for their respective distributions.
For gamma distribution parametrization II:
- If $X_{k} \sim Γ (α_{k}, β_{k})$ are independent, then $\frac{X_{1}}{X_{2}} \sim β^{'} (α_{1}, α_{2}, 1, \frac{β_{2}}{β_{1}})$ . The $β_{k}$ are rate parameters, while $\frac{β_{2}}{β_{1}}$ is a scale parameter.
- If $β_{2} \sim Γ (α_{1}, β_{1})$ and $X_{2} ∣ β_{2} \sim Γ (α_{2}, β_{2})$ , then $X_{2} \sim β^{'} (α_{2}, α_{1}, 1, β_{1})$ . The $β_{k}$ are rate parameters for the gamma distributions, but $β_{1}$ is the scale parameter for the beta prime.
$β^{'} (p, 1, a, b) = Dagum (p, a, b)$ the Dagum distribution
$β^{'} (1, p, a, b) = SinghMaddala (p, a, b)$ the Singh–Maddala distribution.
$β^{'} (1, 1, γ, σ) = LL (γ, σ)$ 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 $α$ , then $\frac{X}{x_{m}} - 1 \sim β^{'} (1, α)$ .
If X has a Lomax distribution, also known as a Pareto Type II distribution, with shape parameter $α$ and scale parameter $λ$ , then $\frac{X}{λ} \sim β^{'} (1, α)$ .
If X has a standard Pareto Type IV distribution with shape parameter $α$ and inequality parameter $γ$ , then $X^{\frac{1}{γ}} \sim β^{'} (1, α)$ , or equivalently, $X \sim β^{'} (1, α, \frac{1}{γ}, 1)$ .
The inverted Dirichlet distribution is a generalization of the beta prime distribution.
If $X \sim β^{'} (α, β)$ , then $\ln X$ has a generalized logistic distribution. More generally, if $X \sim β^{'} (α, β, p, q)$ , then $\ln X$ has a scaled and shifted generalized logistic distribution.
If $X \sim β^{'} (\frac{1}{2}, \frac{1}{2})$ , then $\pm \sqrt{X}$ follows a Cauchy distribution, which is equivalent to a student-t distribution with the degrees of freedom of 1.

Notes

1 2 Johnson et al (1995), p 248
↑ Bourguignon, M.; Santos-Neto, M.; de Castro, M. (2021). "A new regression model for positive random variables with skewed and long tail". Metron. 79: 33–55. doi:10.1007/s40300-021-00203-y. S2CID 233534544.
↑ Dubey, Satya D. (December 1970). "Compound gamma, beta and F distributions". Metrika. 16: 27–31. doi:10.1007/BF02613934. S2CID 123366328.

References

Johnson, N.L., Kotz, S., Balakrishnan, N. (1995). Continuous Univariate Distributions, Volume 2 (2nd Edition), Wiley. ISBN 0-471-58494-0
Bourguignon, M.; Santos-Neto, M.; de Castro, M. (2021), "A new regression model for positive random variables with skewed and long tail", Metron, 79: 33–55, doi:10.1007/s40300-021-00203-y, S2CID 233534544

MathWorld article

[Johnson_et_al_1995,_p_248-1] 1 2 Johnson et al (1995), p 248

[Bourguignon-2] Bourguignon, M.; Santos-Neto, M.; de Castro, M. (2021). "A new regression model for positive random variables with skewed and long tail". Metron. 79: 33–55. doi:10.1007/s40300-021-00203-y. S2CID 233534544.

[Dubey-3] Dubey, Satya D. (December 1970). "Compound gamma, beta and F distributions". Metrika. 16: 27–31. doi:10.1007/BF02613934. S2CID 123366328.

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