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Beta prime distribution
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<h2 id="definitions">Definitions</h2> <p>Beta prime distribution is defined for x > 0 {\displaystyle x>0} with two parameters <i>α</i> and <i>β</i>, having the <a href="/facts/Probability_density_function/zvfybna4">probability density function</a>: </p> f ( x ) = x α − 1 ( 1 + x ) − α − β B ( α , β ) {\displaystyle f(x)={\frac {x^{\alpha -1}(1+x)^{-\alpha -\beta }}{\mathrm {B} (\alpha ,\beta )}}} <p>where <i>B</i> is the <a href="/facts/Beta_function/Dcm1JAxb">Beta function</a>. </p><p>The <a href="/facts/Cumulative_distribution_function/WaKU8tp4">cumulative distribution function</a> is </p> F ( x ; α , β ) = I x 1 + x ( α , β ) , {\displaystyle F(x;\alpha ,\beta )=I_{\frac {x}{1+x}}\left(\alpha ,\beta \right),} <p>where <i>I</i> is the <a href="/facts/Regularized_incomplete_beta_function/Dcm1JAxb">regularized incomplete beta function</a>. </p><p>While the related <a href="/facts/Beta_distribution/DbJk4eeV">beta distribution</a> is the <a href="/facts/Conjugate_prior_distribution/ScCFcs8b">conjugate prior distribution</a> 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 <a href="/facts/Odds/VtM45sI6">odds</a>. The distribution is a <a href="/facts/Pearson_distribution/L0oULaWK">Pearson type VI</a> distribution.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p><p>The mode of a variate <i>X</i> distributed as β ′ ( α , β ) {\displaystyle \beta '(\alpha ,\beta )} is X ^ = α − 1 β + 1 {\displaystyle {\hat {X}}={\frac {\alpha -1}{\beta +1}}} . Its mean is α β − 1 {\displaystyle {\frac {\alpha }{\beta -1}}} if β > 1 {\displaystyle \beta >1} (if β ≤ 1 {\displaystyle \beta \leq 1} the mean is infinite, in other words it has no well defined mean) and its variance is α ( α + β − 1 ) ( β − 2 ) ( β − 1 ) 2 {\displaystyle {\frac {\alpha (\alpha +\beta -1)}{(\beta -2)(\beta -1)^{2}}}} if β > 2 {\displaystyle \beta >2} . </p><p>For − α < k < β {\displaystyle -\alpha <k<\beta } , the <i>k</i>-th moment E [ X k ] {\displaystyle E[X^{k}]} is given by </p> E [ X k ] = B ( α + k , β − k ) B ( α , β ) . {\displaystyle E[X^{k}]={\frac {\mathrm {B} (\alpha +k,\beta -k)}{\mathrm {B} (\alpha ,\beta )}}.} <p>For k ∈ N {\displaystyle k\in \mathbb {N} } with k < β , {\displaystyle k<\beta ,} this simplifies to </p> E [ X k ] = ∏ i = 1 k α + i − 1 β − i . {\displaystyle E[X^{k}]=\prod _{i=1}^{k}{\frac {\alpha +i-1}{\beta -i}}.} <p>The cdf can also be written as </p> x α ⋅ 2 F 1 ( α , α + β , α + 1 , − x ) α ⋅ B ( α , β ) {\displaystyle {\frac {x^{\alpha }\cdot {}_{2}F_{1}(\alpha ,\alpha +\beta ,\alpha +1,-x)}{\alpha \cdot \mathrm {B} (\alpha ,\beta )}}} <p>where 2 F 1 {\displaystyle {}_{2}F_{1}} is the Gauss's <a href="/facts/Hypergeometric_function/yTEC6NBJ">hypergeometric function</a> 2<i>F</i>1 . </p> <h3>Alternative parameterization</h3> <p>The beta prime distribution may also be reparameterized in terms of its mean <i>μ</i> > 0 and precision <i>ν</i> > 0 parameters (<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> p. 36). </p><p>Consider the parameterization <i>μ</i> = <i>α</i>/(<i>β</i> − 1) and <i>ν</i> = <i>β</i> − 2, i.e., <i>α</i> = <i>μ</i>(1 + <i>ν</i>) and <i>β</i> = 2 + <i>ν</i>. Under this parameterization E[<i>Y</i>] = <i>μ</i> and Var[Y] = <i>μ</i>(1 + <i>μ</i>)/<i>ν</i>. </p> <h3>Generalization</h3> <p>Two more parameters can be added to form the generalized beta prime distribution β ′ ( α , β , p , q ) {\displaystyle \beta '(\alpha ,\beta ,p,q)} : </p> <ul><li> p > 0 {\displaystyle p>0} <a href="/facts/Shape_parameter/7jaG8RDp">shape</a> (<a href="/facts/Real_number/R02gw5Pb">real</a>)</li> <li> q > 0 {\displaystyle q>0} scale (<a href="/facts/Real_number/R02gw5Pb">real</a>)</li></ul> <p>having the <a href="/facts/Probability_density_function/zvfybna4">probability density function</a>: </p> f ( x ; α , β , p , q ) = p ( x q ) α p − 1 ( 1 + ( x q ) p ) − α − β q B ( α , β ) {\displaystyle 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 }}{q\mathrm {B} (\alpha ,\beta )}}} <p>with <a href="/facts/Mean/swcEd4Pg">mean</a> </p> q Γ ( α + 1 p ) Γ ( β − 1 p ) Γ ( α ) Γ ( β ) if β p > 1 {\displaystyle {\frac {q\Gamma \left(\alpha +{\tfrac {1}{p}}\right)\Gamma (\beta -{\tfrac {1}{p}})}{\Gamma (\alpha )\Gamma (\beta )}}\quad {\text{if }}\beta p>1} <p>and <a href="/facts/Mode_(statistics)/vq5tGp0G">mode</a> </p> q ( α p − 1 β p + 1 ) 1 p if α p ≥ 1 {\displaystyle q\left({\frac {\alpha p-1}{\beta p+1}}\right)^{\tfrac {1}{p}}\quad {\text{if }}\alpha p\geq 1} <p>Note that if <i>p</i> = <i>q</i> = 1 then the generalized beta prime distribution reduces to the standard beta prime distribution. </p><p>This generalization can be obtained via the following invertible transformation. If y ∼ β ′ ( α , β ) {\displaystyle y\sim \beta '(\alpha ,\beta )} and x = q y 1 / p {\displaystyle x=qy^{1/p}} for q , p > 0 {\displaystyle q,p>0} , then x ∼ β ′ ( α , β , p , q ) {\displaystyle x\sim \beta '(\alpha ,\beta ,p,q)} . </p> <h4>Compound gamma distribution</h4> <p>The compound gamma distribution<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> is the generalization of the beta prime when the scale parameter, <i>q</i> is added, but where <i>p</i> = 1. It is so named because it is formed by <a href="/facts/Compound_distribution/ccJIMBxT">compounding</a> two <a href="/facts/Gamma_distribution/lczcdJmw">gamma distributions</a>: </p> β ′ ( x ; α , β , 1 , q ) = ∫ 0 ∞ G ( x ; α , r ) G ( r ; β , q ) d r {\displaystyle \beta '(x;\alpha ,\beta ,1,q)=\int _{0}^{\infty }G(x;\alpha ,r)G(r;\beta ,q)\;dr} <p>where G ( x ; a , b ) {\displaystyle G(x;a,b)} is the gamma pdf with shape a {\displaystyle a} and inverse scale b {\displaystyle b} . </p><p>The mode, mean and variance of the compound gamma can be obtained by multiplying the mode and mean in the above infobox by <i>q</i> and the variance by <i>q</i>2. </p><p>Another way to express the compounding is if r ∼ G ( β , q ) {\displaystyle r\sim G(\beta ,q)} and x ∣ r ∼ G ( α , r ) {\displaystyle x\mid r\sim G(\alpha ,r)} , then x ∼ β ′ ( α , β , 1 , q ) {\displaystyle x\sim \beta '(\alpha ,\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. </p> <h2 id="properties">Properties</h2> <ul><li>If X ∼ β ′ ( α , β ) {\displaystyle X\sim \beta '(\alpha ,\beta )} then 1 X ∼ β ′ ( β , α ) {\displaystyle {\tfrac {1}{X}}\sim \beta '(\beta ,\alpha )} .</li> <li>If Y ∼ β ′ ( α , β ) {\displaystyle Y\sim \beta '(\alpha ,\beta )} , and X = q Y 1 / p {\displaystyle X=qY^{1/p}} , then X ∼ β ′ ( α , β , p , q ) {\displaystyle X\sim \beta '(\alpha ,\beta ,p,q)} .</li> <li>If X ∼ β ′ ( α , β , p , q ) {\displaystyle X\sim \beta '(\alpha ,\beta ,p,q)} then k X ∼ β ′ ( α , β , p , k q ) {\displaystyle kX\sim \beta '(\alpha ,\beta ,p,kq)} .</li> <li> β ′ ( α , β , 1 , 1 ) = β ′ ( α , β ) {\displaystyle \beta '(\alpha ,\beta ,1,1)=\beta '(\alpha ,\beta )} </li></ul> <h2 id="related-distributions">Related distributions</h2> <ul><li>If X ∼ Beta ( α , β ) {\displaystyle X\sim {\textrm {Beta}}(\alpha ,\beta )} , then X 1 − X ∼ β ′ ( α , β ) {\displaystyle {\frac {X}{1-X}}\sim \beta '(\alpha ,\beta )} . This property can be used to generate beta prime distributed variates.</li> <li>If X ∼ β ′ ( α , β ) {\displaystyle X\sim \beta '(\alpha ,\beta )} , then X 1 + X ∼ Beta ( α , β ) {\displaystyle {\frac {X}{1+X}}\sim {\textrm {Beta}}(\alpha ,\beta )} . This is a corollary from the property above.</li> <li>If X ∼ F ( 2 α , 2 β ) {\displaystyle X\sim F(2\alpha ,2\beta )} has an <a href="/facts/F-distribution/XNuuJtWY"><i>F</i>-distribution</a>, then α β X ∼ β ′ ( α , β ) {\displaystyle {\tfrac {\alpha }{\beta }}X\sim \beta '(\alpha ,\beta )} , or equivalently, X ∼ β ′ ( α , β , 1 , β α ) {\displaystyle X\sim \beta '(\alpha ,\beta ,1,{\tfrac {\beta }{\alpha }})} .</li> <li>For <a href="/facts/Gamma_distribution/lczcdJmw">gamma distribution</a> parametrization I: <ul><li>If X k ∼ Γ ( α k , θ k ) {\displaystyle X_{k}\sim \Gamma (\alpha _{k},\theta _{k})} are independent, then X 1 X 2 ∼ β ′ ( α 1 , α 2 , 1 , θ 1 θ 2 ) {\displaystyle {\tfrac {X_{1}}{X_{2}}}\sim \beta '(\alpha _{1},\alpha _{2},1,{\tfrac {\theta _{1}}{\theta _{2}}})} . Note θ 1 , θ 2 , θ 1 θ 2 {\displaystyle \theta _{1},\theta _{2},{\tfrac {\theta _{1}}{\theta _{2}}}} are all scale parameters for their respective distributions.</li></ul></li> <li>For gamma distribution parametrization II: <ul><li>If X k ∼ Γ ( α k , β k ) {\displaystyle X_{k}\sim \Gamma (\alpha _{k},\beta _{k})} are independent, then X 1 X 2 ∼ β ′ ( α 1 , α 2 , 1 , β 2 β 1 ) {\displaystyle {\tfrac {X_{1}}{X_{2}}}\sim \beta '(\alpha _{1},\alpha _{2},1,{\tfrac {\beta _{2}}{\beta _{1}}})} . The β k {\displaystyle \beta _{k}} are rate parameters, while β 2 β 1 {\displaystyle {\tfrac {\beta _{2}}{\beta _{1}}}} is a scale parameter.</li> <li>If β 2 ∼ Γ ( α 1 , β 1 ) {\displaystyle \beta _{2}\sim \Gamma (\alpha _{1},\beta _{1})} and X 2 ∣ β 2 ∼ Γ ( α 2 , β 2 ) {\displaystyle X_{2}\mid \beta _{2}\sim \Gamma (\alpha _{2},\beta _{2})} , then X 2 ∼ β ′ ( α 2 , α 1 , 1 , β 1 ) {\displaystyle X_{2}\sim \beta '(\alpha _{2},\alpha _{1},1,\beta _{1})} . The β k {\displaystyle \beta _{k}} are rate parameters for the gamma distributions, but β 1 {\displaystyle \beta _{1}} is the scale parameter for the beta prime.</li></ul></li> <li> β ′ ( p , 1 , a , b ) = Dagum ( p , a , b ) {\displaystyle \beta '(p,1,a,b)={\textrm {Dagum}}(p,a,b)} the <a href="/facts/Dagum_distribution/rmgoUXvM">Dagum distribution</a></li> <li> β ′ ( 1 , p , a , b ) = SinghMaddala ( p , a , b ) {\displaystyle \beta '(1,p,a,b)={\textrm {SinghMaddala}}(p,a,b)} the <a href="/facts/Singh%25E2%2580%2593Maddala_distribution/75pvKu6P">Singh–Maddala distribution</a>.</li> <li> β ′ ( 1 , 1 , γ , σ ) = LL ( γ , σ ) {\displaystyle \beta '(1,1,\gamma ,\sigma )={\textrm {LL}}(\gamma ,\sigma )} the <a href="/facts/Log_logistic_distribution/ze7Pasb3">log logistic distribution</a>.</li> <li>The beta prime distribution is a special case of the type 6 <a href="/facts/Pearson_distribution/L0oULaWK">Pearson distribution</a>.</li> <li>If <i>X</i> has a <a href="/facts/Pareto_distribution/PgMDbLrR">Pareto distribution</a> with minimum x m {\displaystyle x_{m}} and shape parameter α {\displaystyle \alpha } , then X x m − 1 ∼ β ′ ( 1 , α ) {\displaystyle {\dfrac {X}{x_{m}}}-1\sim \beta ^{\prime }(1,\alpha )} .</li> <li>If <i>X</i> has a <a href="/facts/Lomax_distribution/ch9VY1mn">Lomax distribution</a>, also known as a Pareto Type II distribution, with shape parameter α {\displaystyle \alpha } and scale parameter λ {\displaystyle \lambda } , then X λ ∼ β ′ ( 1 , α ) {\displaystyle {\frac {X}{\lambda }}\sim \beta ^{\prime }(1,\alpha )} .</li> <li>If <i>X</i> has a standard <a href="/facts/Pareto_distribution/PgMDbLrR">Pareto Type IV distribution</a> with shape parameter α {\displaystyle \alpha } and inequality parameter γ {\displaystyle \gamma } , then X 1 γ ∼ β ′ ( 1 , α ) {\displaystyle X^{\frac {1}{\gamma }}\sim \beta ^{\prime }(1,\alpha )} , or equivalently, X ∼ β ′ ( 1 , α , 1 γ , 1 ) {\displaystyle X\sim \beta ^{\prime }(1,\alpha ,{\tfrac {1}{\gamma }},1)} .</li> <li>The <a href="/facts/Inverted_Dirichlet_distribution/qH7e3uAT">inverted Dirichlet distribution</a> is a generalization of the beta prime distribution.</li> <li>If X ∼ β ′ ( α , β ) {\displaystyle X\sim \beta '(\alpha ,\beta )} , then ln X {\displaystyle \ln X} has a <a href="/facts/Generalized_logistic_distribution/bH0v6w4n">generalized logistic distribution</a>. More generally, if X ∼ β ′ ( α , β , p , q ) {\displaystyle X\sim \beta '(\alpha ,\beta ,p,q)} , then ln X {\displaystyle \ln X} has a <a href="/facts/Sigmoid-beta_distribution/bH0v6w4n">scaled and shifted</a> generalized logistic distribution.</li> <li>If X ∼ β ′ ( 1 2 , 1 2 ) {\displaystyle X\sim \beta '\left({\frac {1}{2}},{\frac {1}{2}}\right)} , then ± X {\displaystyle \pm {\sqrt {X}}} follows a Cauchy distribution, which is equivalent to a student-t distribution with the degrees of freedom of 1.</li></ul> <h2 id="notes">Notes</h2> <ul><li>Johnson, N.L., Kotz, S., Balakrishnan, N. (1995). <i>Continuous Univariate Distributions</i>, Volume 2 (2nd Edition), Wiley. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-58494-0</li> <li>Bourguignon, M.; Santos-Neto, M.; de Castro, M. (2021), "A new regression model for positive random variables with skewed and long tail", <i>Metron</i>, 79: 33–55, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fs40300-021-00203-y">10.1007/s40300-021-00203-y</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:233534544">233534544</a></li></ul> <ul><li><a href="http://mathworld.wolfram.com/BetaPrimeDistribution.html">MathWorld article</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Johnson et al (1995), p 248 <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Johnson et al (1995), p 248 <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>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. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Dubey, Satya D. (December 1970). "Compound gamma, beta and F distributions". Metrika. 16: 27–31. doi:10.1007/BF02613934. S2CID 123366328. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> </ol>