The Amoroso Distribution

Transcription

1 The Amoroso Distribution Gavin E. Crooks Physical Biosciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 9470 Abstract: The Amoroso distribution is the natural unification of the gamma and extreme value families of distributions. Over 50 distinct, named distributions (and twice as many synonyms) occur as special cases or limiting forms. Consequently, this single simple functional form encapsulates and systematizes an extensive menagerie of interesting and common probability distributions. Tech. Note 003v6 (00-0-) Contents The Amoroso distribution family Special cases: Miscellaneous Special cases: Positive integer β Special cases: Negative integer β Special cases: Extreme order statistics Properties Log-gamma distributions Special cases Properties Miscellaneous limits A Index of distributions References The Amoroso distribution family The Amoroso (generalized gamma, Stacy-Mihram) distribution [, ] is a four parameter, continuous, univariate, unimodal probability density, with semiinfinite range. The functional form in the most straightforward parameterization is Amoroso(x a,, α, β) = Γ(α) ( ) { β αβ ( ) } β x a x a exp for x, a,, α, β in R, α > 0, support x a if > 0, x a if < 0. The Amoroso distribution was originally developed to model lifetimes []. It occurs as the Weibullization of the standard gamma distribution (5) and, with ()

2 / integer α, in extreme value statistics (8). The Amoroso distribution is itself a limiting form of various more general distributions, most notable the generalized beta and generalized beta prime distributions [3]. A useful and important property of the Amoroso distribution is that many common and interesting probability distributions are special cases or limiting forms (See Table ). (Informally, an interesting distribution is one that has acquired a name, which generally indicates that the distribution is the solution to one or more interesting problems.) This provides a convenient method for systemizing a significant fraction of the probability distributions that are encountered in practice, provides a consistent parameterization for those distributions, and to obviates the need to enumerate the properties (mean, mode, variance, entropy and so on) of each and every specialization. Notation: The four real parameters of the Amoroso distribution consist of a location parameter a, a scale parameter, and two shape parameters, α and β. Whenever these symbols appears in special cases or limiting forms, they refer directly to the parameters of the Amoroso distribution. The shape parameter α is positive, and in many specializations an integer, α = n, or half-integer, α = k. The negation of a standard parameter is indicated by a bar, e.g. β = β. The chi, chi-squared and related distributions are traditionally parameterized with the scale parameter σ, where = (σ ) /β, and σ is the standard deviation of an underlying normal distribution. Additional alternative parameters are introduced as necessary. We write Amoroso(x a,, α, β) for a density function, Amoroso(a,, α, β) for the corresponding random variables, and X Amoroso(a,, α, β) to indicate that two random variables have the same probability distribution [5]. Recall that a random variable is an unbound function from events to values, whereas the probability density function maps from values to probabilities... Special cases: Miscellaneous Stacy (hyper gamma, generalized Weibull, Nukiyama-Tanasawa, generalized gamma, generalized semi-normal, hydrograph, Leonard hydrograph, transformed gamma) distribution [4, 6]: Stacy(x, α, β) = Γ(α) β ( x =Amoroso(x 0,, α, β) ) { αβ ( x ) } β exp If we drop the location parameter from Amoroso, then we obtain the Stacy, or generalized gamma distribution, the parent of the gamma family of distributions. If β is negative then the distribution is generalized inverse gamma, the parent of various inverse distributions, including the inverse gamma (0) and inverse chi (6). ()

3 / 3 Table The Amoroso family of distributions. Amoroso(x a,, α, β) = β ( ) { x a αβ ( ) } x a β exp Γ(α) for x, a,, α, β in R, α > 0, k, n positive integers support x a if > 0, x a if < 0. () Amoroso a α β () Stacy 0... (8) gen. Fisher-Tippett.. n. (9) Fisher-Tippett... (33) Fréchet.. <0 (3) generalized Fréchet.. n <0 (5) scaled inverse chi 0. k - (6) inverse chi 0 k - (7) inverse Rayleigh 0. - (9) Pearson type V... - (0) inverse gamma (3) scaled inverse chi-square 0. k - (4) inverse chi-square 0 k - () Lévy.. - () inverse exponential 0. - (7) Pearson type III... (5) gamma 0.. (5) Erlang 0 >0 n (6) standard gamma 0. (3) scaled chi-square 0. k () chi-square 0 k (9) shifted exponential.. (8) exponential 0. (8) standard exponential 0 (5) Wien 0. 4 (0) Nakagami... (5) scaled chi 0. k (4) chi 0 k () half-normal 0. (6) Rayleigh 0. 3 (7) Maxwell 0. (8) Wilson-Hilferty (30) generalized Weibull.. n >0 (3) Weibull.. >0 (4) pseudo-weibull.. + β >0 (3) stretched exponential 0. >0 Limits (34) log-gamma.... lim β p (44) power law... lim β β 0 (4) log-normal.. (βσ). lim β 0 (43) normal... lim α

4 / 4.5 β=5 β= β= β=4 β= Fig. Amoroso(x 0,,, β), stretched exponential The Stacy distribution is obtained as the positive even powers, modulus, and powers of the modulus of a centered, normal random variable (43), ) Stacy ((σ ) β,, β β Normal(0, σ) and as powers of the sum of squares of k centered, normal random variables. ) Stacy ((σ ) β, k, β ( k i= ) ( ) β Normal(0, σ) Stretched exponential distribution [7]: StretchedExp(x, β) = β ( x ) { β ( x ) } β exp for β > 0 =Weibull(x 0,, β) =Amoroso(x 0,,, β) Stretched exponentials are an alternative to power laws for modeling fat tailed distributions. For β = we recover the exponential distribution (8), and β = 0 a power law distribution (44). (3) Pseudo-Weibull distribution [44]: β ( x ) { β ( x ) } β PseudoWeibull(x, β) = exp Γ( + β ) for β > 0 =Amoroso(x 0,, + β, β) (4)

5 / 5.5 β=4 β=3, Wilson-Hilferty 0.5 β=, scaled chi β=, gamma Fig. Amoroso(x 0,,, β) Proposed as another model of failure times... Special cases: Positive integer β Gamma (Γ) distribution [35, 36, ] : ( x ) α { Gamma(x, α) = exp x } Γ(α) = PearsonIII(x 0,, α) = Stacy(x, α, ) = Amoroso(x 0,, α, ) (5) The name of this distribution derives from the normalization constant. The gamma distribution often appear as a solution to problems in statistical physics. For example, the energy density of a classical ideal gas, or the Wien (Vienna) distribution Wien(x T ) = Gamma(x T, 4), is an approximation to the relative intensity of black body radiation as a function of the frequency. The Erlang (m- Erlang) distribution [8] is a gamma distribution with integer α, which models the waiting time to observe α events from a Poisson process with rate / ( > 0). Gamma distributions obey an addition property: Gamma(, α ) + Gamma(, α ) Gamma(, α + α ) The sum of two independent, gamma distributed random variables (with common s, but possibly different α s) is again a gamma random variable []. Standard gamma (standard Amoroso) distribution []: StdGamma(x α) = Γ(α) xα e x (6)

6 / 6.5 α= α=8 α=6 α=4 α= Fig 3. Gamma(x, α) (unit variance) α The Amoroso distribution can be obtained from the standard gamma distribution by the Weibull change of variables, x ( ) x a β. [ ] /β Amoroso(a,, α, β) a + StdGamma(α) Pearson type III distribution [36, ]: PearsonIII(x a,, α) = Γ(α) ( ) α { ( )} x a x a exp =Amoroso(x a,, α, ) (7) The gamma distribution with a location parameter. Exponential (Pearson type X, waiting time, negative exponential) distribution []: Exp(x ) = { exp x } = Gamma(x, ) = Amoroso(x 0,,, ) An important property of the exponential distribution is that it is memoryless: the conditional probability given that x > c, where c is a positive content, is again an exponential distribution with the same scale parameter. The only other distribution with this property is the geometric distribution [9], the discrete analog of the exponential distribution. With = we obtain a standard exponential distribution. See also shifted exponential (9), stretched exponential (3) and inverse exponential (). (8)

7 / 7 Shifted exponential distribution []: ShiftExp(x a, ) = { ( )} x a exp = PearsonIII(x a,, ) = Amoroso(x a,,, ) (9) The exponential distribution with a location parameter. Nakagami (generalized normal, Nakagami-m) distribution [34]: ( ) { m ( ) } x a x a Nakagami(x a,, m) = Γ( m ) exp = Amoroso(x a,, m, ) (0) Used to model attenuation of radio signals that reach a receiver by multiple paths [34]. Half-normal (semi-normal, positive definite normal, one-sided normal) distribution []: ( )} x HalfNormal(x σ) = { exp πσ σ () = ScaledChi(x σ, ) = Stacy(x σ,, ) = Amoroso(x 0, σ,, ) The modulus of a normal distribution with zero mean and variance σ. Chi-square (χ ) distribution [, ]: ChiSqr(x k) = ( x ) k { ( x )} Γ( k ) exp for positive integer k = Gamma(x, k ) = Stacy(x, k, ) = Amoroso(x 0,, k, ) () The distribution of a sum of squares of k independent standard normal random variables. The chi-square distribution is important for statistical hypothesis testing in the frequentist approach to statistical inference.

8 / k= k= k=3 k=4 k= Fig 4. ChiSqr(x k) Scaled chi-square distribution [8]: ScaledChiSqr(x σ, k) = ( x ) k { ( x )} exp σ Γ( k ) σ σ for positive integer k = Stacy(x σ, k, ) = Gamma(x σ, k ) = Amoroso(x 0, σ, k, ) (3) The distribution of a sum of squares of k independent normal random variables with variance σ. Chi (χ) distribution []: ( ) k { ( )} x x Chi(x k) = Γ( k ) exp for positive integer k = ScaledChi(x, k) = Stacy(x, k, ) = Amoroso(x 0,, k, ) (4) The root-mean-square of k independent standard normal variables, or the square root of a chi-square random variable. Chi(k) ChiSqr(k)

9 / 9.5 α=/, half-normal α=, Rayleigh α=3/, Maxwell Fig 5. Amoroso(x 0,, α, ) Scaled chi (generalized Rayleigh) distribution [33, ]: ( ) k { ( )} x x ScaledChi(x σ, k) = Γ( k ) exp σ σ σ for positive integer k = Stacy(x σ, k, ) (5) = Amoroso(x 0, σ, k, ) The root-mean-square of k independent and identically distributed normal variables with zero mean and variance σ. Rayleigh distribution [4, ]: Rayleigh(x σ) = ( )} x { σ x exp σ = ScaledChi(x σ, ) (6) = Stacy(x σ,, ) = Amoroso(x 0, σ,, ) The root-mean-square of two independent and identically distributed normal variables with zero mean and variance σ. For instance, wind speeds are approximately Rayleigh distributed, since the horizontal components of the velocity are approximately normal, and the vertical component is typically small [4].

10 / 0 Maxwell (Maxwell-Boltzmann, Maxwell speed) distribution [30, ]: { ( )} x Maxwell(x σ) = πσ 3 x exp σ = ScaledChi(x σ, 3) = Stacy(x σ, 3, ) = Amoroso(x 0, σ, 3, ) (7) The speed distribution of molecules in thermal equilibrium. The root-meansquare of three independent and identically distributed normal variables with zero mean and variance σ. Wilson-Hilferty distribution [47, ]: WilsonHilferty(x, α) = 3 ( x ) { 3α ( x ) } 3 exp Γ(α) = Stacy(x, α, 3) = Amoroso(x 0,, α, 3) (8) The cube root of a gamma variable follows the Wilson-Hilferty distribution [47], which has been used to approximate a normal distribution if α is not too small. A related approximation using quartic roots of gamma variables [9] leads to Amoroso(x 0,, α, 4)..3. Special cases: Negative integer β Pearson type V distribution [37]: PearsonV(x a,, α) = ( ) α+ { ( )} exp Γ(α) x a x a =Amoroso(x a,, α, ) (9) With negative β we obtain various inverse distributions related to distributions with positive β by the reciprocal transformation ( x a ) ( x a ). Pearson s type V is the inverse of Pearson s type III distribution. Inverse gamma (Vinci) distribution []: InvGamma(x, α) = Γ(α) ( ) α+ { exp x = Stacy(x, α, ) = PearsonV(x 0,, α) = Amoroso(x 0,, α, ) ( )} x (0)

11 /.5.5 β=-3 β=- scaled β=- inverse-chi inverse gamma Fig 6. Amoroso(x 0,,, β), negative β. Occurs as the conjugate prior for an exponential distribution s scale parameter [], or the prior for variance of a normal distribution with known mean [5]. Inverse exponential distribution [5]: InvExp(x ) = x exp { ( )} x = InvGamma(x, ) = Stacy(x,, ) = Amoroso(x 0,,, ) () Note that the name inverse exponential is occasionally used for the ordinary exponential distribution (8). Lévy distribution (van der Waals profile) [0]: { } c c Lévy(x a, c) = exp π (x a) 3/ (x a) = PearsonV(x a, c, ) = Amoroso(x a, c,, ) () The Lévy distribution is notable for being stable: a linear combination of identically distributed Lévy distributions is again a Lévy distribution. The other stable distributions with analytic forms are the normal distribution (43), which is also a limit of the Amoroso distribution, and the Cauchy distribution [], which is not. Lévy distributions describe first passage times in one dimensional Brownian diffusion [0].

12 Scaled inverse chi-square distribution [5]: / ( ) k ScaledInvChiSqr(x σ, k) = σ + { ( )} Γ( k ) σ exp x σ x for positive integer k = InvGamma(x σ, k ) = PearsonV(x 0, σ, k ) = Stacy(x σ, k, ) = Amoroso(x 0, σ, k, ) (3) A special case of the inverse gamma distribution with half-integer α. Used as a prior for variance parameters in normal models [5]. Inverse chi-square distribution [5]: InvChiSqr(x k) = ( ) k + { ( )} Γ( k ) exp x x for positive integer k = ScaledInvChiSqr(x, k) = InvGamma(x, k ) = PearsonV(x 0,, k ) = Stacy(x, k, ) = Amoroso(x 0,, k, ) (4) A standard scaled inverse chi-square distribution. Scaled inverse chi distribution [8]: ScaledInvChi(x σ, k) = σ Γ( k ) ( k+ { ( )} exp σ x) σ x = Stacy(x σ, k, ) = Amoroso(x 0, σ, k, ) (5) Used as a prior for the standard deviation of a normal distribution. Inverse chi distribution [8]: InvChi(x k) = ( ) k+ { ( )} Γ( k ) exp x x = Stacy(x, k, ) = Amoroso(x 0,, k, ) (6)

13 / 3 standard Gumbel reversed Weibull, β= Frechet, β= Fig 7. Extreme value distributions The standard inverse chi distribution. Inverse Rayleigh distribution [9]: InvRayleigh(x σ) = ( 3 { ( )} σ exp σ x) σ x = Stacy(x σ,, ) = Amoroso(x 0, σ,, ) (7) The inverse Rayleigh distribution has been used to model a failure time [43]..4. Special cases: Extreme order statistics Generalized Fisher-Tippett distribution [40, 3]: ( ) { GenFisherTippett(x a, ω, n, β) = nn β nβ ( ) } β x a x a Γ(n) ω exp n ω ω for positive integer n (8) = Amoroso(x a, ω/n β, n, β) If we take N samples from a probability distribution, then asymptotically for large N and n N, the distribution of the nth largest (or smallest) sample follows a generalized Fisher-Tippett distribution. The parameter β depends on the tail behavior of the sampled distribution. Roughly speaking, if the tail is unbounded and decays exponentially then β limits to, if the tail scales as a power law then β < 0, and if the tail is finite β > 0 [7]. In these three limits we obtain the Gumbel (39, 38), Fréchet (33, 3) and Weibull (3,30) families

14 / 4 of extreme value distribution (Extreme value distributions types I, II and III) respectively. If β/ω is negative we obtain distributions for the nth maxima, if positive then the nth minima. Fisher-Tippett (Generalized extreme value, GEV, von Mises-Jenkinson, von Mises extreme value) distribution [, 45, 7, 3]: ( ) { FisherTippett(x a, ω, β) = β β ( ) } β x a x a ω exp (9) ω ω = GenFisherTippett(x a, ω,, β) = Amoroso(x a, ω,, β) The asymptotic distribution of the extreme value from a large sample. The superclass of type I, II and III (Gumbel, Fréchet, Weibull) extreme value distributions [45]. This is the distribution for maximum values with β/ω < 0 and minimum values for β/ω > 0. The maximum of two Fisher-Tippett random variables (minimum if β/ω > 0) is again a Fisher-Tippett random variable. [ ] max FisherTippett(a, ω, β), FisherTippett(a, ω, β) FisherTippett(a, (ωβ + ωβ )/β ω ω, β) This follows because taking the maximum of two random variables is equivalent to multiplying their cumulative distribution { functions, and the Fisher-Tippett cumulative distribution function is exp ( ) } x a β ω. Generalized Weibull distribution [40, 3]: ( ) { nβ ( ) } β GenWeibull(x a, ω, n, β) = nn β x a x a exp n Γ(n) ω ω ω for β > 0 = GenFisherTippett(x a, ω, n, β) = Amoroso(x a, ω/n β, n, β) (30) The limiting distribution of the nth smallest value of a large number of identically distributed random variables that are at least a. If ω is negative we obtain the distribution of the nth largest value. Weibull (Fisher-Tippett type III, Gumbel type III, Rosin-Rammler, Rosin- Rammler-Weibull, extreme value type III, Weibull-Gnedenko) distribution [46,

15 / 5 3]: Weibull(x a, ω, β) = β ω ( ) { β ( ) } β x a x a exp ω ω for β > 0 = FisherTippett(x a, ω, β) = Amoroso(x a, ω,, β) (3) This is the limiting distribution of the minimum of a large number of identically distributed random variables that are at least a. If ω is negative we obtain a reversed Weibull (extreme value type III) distribution for maxima. Special cases of the Weibull distribution include the exponential (β = ) and Rayleigh (β = ) distributions. Generalized Fréchet distribution [40, 3]: ( ) { n β ( GenFréchet(x a, ω, n, β) = nn β x a x a exp n Γ(n) ω ω ω ) β} for β > 0 (3) = GenFisherTippett(x a, ω, n, β) = Amoroso(x a, ω/n β, n, β), The limiting distribution of the nth largest value of a large number identically distributed random variables whose moments are not all finite and are bounded from below by a. (If the shape parameter ω is negative then minimum rather than maxima.) Fréchet (extreme value type II, Fisher-Tippett type II, Gumbel type II, inverse Weibull) distribution [3, 7]: Fréchet(x a, ω, β) = β ω ( ) { β ( x a x a exp ω ω for β > 0 = FisherTippett(x a, ω, β) = Amoroso(x a, ω,, β) ) β} (33) The limiting distribution of the largest of a large number identically distributed random variables whose moments are not all finite and are bounded from below by a. (If the shape parameter ω is negative then minimum rather than maxima.) Special cases of the Fréchet distribution include the inverse exponential ( β = ) and inverse Rayleigh ( β = ) distributions.

16 / 6.5. Properties support x a > 0 cdf std. moments x a < 0 Q(α, ( ) x a β) β > 0 Q(α, ( ) x a β) β < 0 mode a + (α β ) β αβ a αβ Γ(α + r β ) Γ(α) mean a + Γ(α + β ) Γ(α) [ Γ(α + variance β ) Γ(α + β ) Γ(α) Γ(α) ) entropy ln Γ(α) β (a = 0, = ) α + n β 0 + α + ( β α ] α + β 0 α + β 0 ψ(α) [6] Here, cdf is the cumulative distribution function, Q(α, x) = Γ(α, x)/γ(α) is the regularized gamma function [], Γ(α, x) = x tα e t dt is the incomplete gamma integral [], and ψ(x) = d dx ln Γ(x) is the digamma function [], the logarithmic derivative of the gamma function. Important special cases and limits include Γ( ) = π, Γ(, x) = πerfc( x) and Γ(, x) = exp( x). The derivative of the regularized gamma function is d dxq(α, x) = Γ(α) xα e x. The profile of the Amoroso distribution is bell shaped for αβ, and otherwise L- or J- shaped with the mode at the boundary. The moments are undefined if the side conditions are not satisfied. Expressions for skew and kurtosis are not simple, but can be deduced from the moments if necessary. The Amoroso distribution can be obtained from the standard gamma distribution (6) with the change of variables, x ( ) x a β. Therefore, Amoroso random numbers can be obtained by sampling from the standard gamma distribution, for instance using the Marsaglia-Tsang fast gamma method [9] and applying the appropriate transformation [6].

17 / 7. Log-gamma distributions The log-gamma (Coale-McNeil) distribution [4, 39, 3] is a three parameter, continuous, univariate, unimodal probability density with infinite range. The functional form in the most straightforward parameterization is { ( ) ( )} x ν x ν LogGamma(x ν, λ, α) = Γ(α) λ exp α exp (34) λ λ for x, ν, λ, α, in R, α > 0, support x The three real parameters consist of a location parameter ν, a scale parameter λ, and a shape parameter α, which is inherited directly from the Amoroso distribution. The name log-gamma arises because the standard log-gamma distribution is the logarithmic transform of the standard gamma distribution ( ) StdLogGamma(α) ln StdGamma(α) ) LogGamma(ν, λ, α) ln (Amoroso(0, e ν, α, λ ) Note that this naming convention is the opposite of that used for the log-normal distribution (4). The name log-gamma has also been used for the antilog transform of the generalized gamma distribution, which leads to the unit-gamma distribution [8]. The log-gamma distribution is a limit of the Amoroso distribution (), and itself has a number of important limits and special cases (Table II), which we will discuss below. LogGamma(x ν, λ, α) (35) ( = lim + ) { αβ ( x ν exp + ) } β x ν β Γ(α) λ β λ β λ = lim Amoroso(ν βλ, βλ, α, β) β Recall that lim β ( + x β )β = exp(x)... Special cases Standard log-gamma distribution: StdLogGamma(x α) = exp {αx exp(x)} (36) Γ(α) =LogGamma(x 0,, α)

18 / 8 Table The log-gamma family of distributions. LogGamma(x ν, λ, α) { ( ) ( )} x ν x ν = Γ(α) λ exp α exp λ λ for x, ν, λ, α, in R, α > 0, support x (34) log-gamma ν λ α (36) standard log-gamma 0 α k (37) log-chi-square ln (38) generalized Gumbel.. n (39) Gumbel.. π (4) BHP.. (40) standard Gumbel 0 - Limits (43) normal.. α lim α The log-gamma distribution with zero location and unit scale. Log-chi-square distribution [8]: { k LogChiSqr(x k) = k Γ( k ) exp x } exp(x) for positive integer k (37) = LogGamma(x ln,, k ) The logarithmic transform of the chi-square distribution (). Generalized Gumbel distribution [7, 3]: { ( ) ( GenGumbel(x u, λ, n) = nn x u Γ(n) λ exp n n exp x u )} λ λ for positive integer n (38) = LogGamma(x u + λ ln n, λ, n) The limiting distribution of the nth largest value of a large number of unbounded identically distributed random variables whose probability distribution has an exponentially decaying tail. Gumbel (Fisher-Tippett type I, Fisher-Tippett-Gumbel, FTG, Gumbel-Fisher- Tippett, log-weibull, extreme value (type I), doubly exponential, double exponential) distribution [, 7, 3]: Gumbel(x u, λ) = λ exp { ( x u = LogGamma(x u, λ, ) λ ) ( exp x u )} λ (39)

19 / 9 α=4 α=5 α=3 0.5 α= α= Fig 8. LogGamma(x 0,, α) This is the asymptotic extreme value distribution for variables of exponential type, unbounded with finite moments [7]. With positive scale λ > 0, this is an extreme value distribution of the maximum, with negative scale λ < 0 (λ > 0) an extreme value distribution of the minimum. Note that the Gumbel is sometimes defined with the negative of the scale used here. Note that the term double exponential distribution can refer to either the Gumbel or Laplace [3] distributions. The Gompertz distribution is a truncated Gumbel distribution [3]. Standard Gumbel (Gumbel) distribution [7]: StdGumbel(x) = exp { x e x} (40) =LogGamma(x 0,, ) The Gumbel distribution with zero location and a unit scale. BHP (Bramwell-Holdsworth-Pinton) distribution [5]: { ( ) ( )} π x ν x ν BHP(x ν, λ) = exp exp ) λ λ λ Γ( π = LogGamma(x ν, λ, π ) (4) Proposed as a model of rare fluctuations in turbulence and other correlated systems.

20 / 0.. Properties support x + cdf cgf mode mean variance Q(α, e x ν λ ) for λ > 0 Q(α, e x ν λ ) for λ < 0 νt + ln ν λ ln α ν + λψ(α) λ ψ (α) Γ(α + λt) Γ(α) skew sgn(λ)ψ (α)/ψ (α) 3/ kurtosis ψ 3 (α)/ψ (α) entropy ln Γ(α) λ αψ(α) + α Here, cdf is the cumulative distribution function, cgf the cumulant generating function, ln E[exp(tX)], sgn(z) is the sign function, ψ n (z) = dn dz ln Γ(z) is the n polygamma function and ψ(z) ψ 0 (z) is the digamma function. [3] 3. Miscellaneous limits Log-normal (Galton, Galton-McAlister, antilog-normal, logarithmic-normal, logarithmico-normal, Cobb-Douglas, Λ) distribution [4, 3, ]: ( ) { ϑ x a LogNormal(x a, ϑ, σ) = exp ( πσ ϑ σ ln x a ) } (4) ϑ = lim β 0 Amoroso(x a, ϑ(βσ) /β, /(βσ), β) The log-normal distribution is a limiting form of the gamma family. To see this, make the requisite substitutions, Amoroso(x a, ϑ(βσ) /β, /(βσ), β) ( ) x a exp ϑ { βσ ln x a ϑ β σ exp [ β ln x a ]} ϑ and in the limit β 0 expand the second exponential to second order in β. With a = 0, ϑ =, σ = we obtain the standard log-normal (Gibrat) distribution [6]. The two-parameter lognormal distribution (a = 0) arises from the multiplicative version of the central limits theorem: When the sum of independent random variables limits to normal, the product of those random

21 / variables limits to log-normal. The log-normal distribution maps to the normal distribution with the transformation x exp(x). ( ) LogNormal(a, ϑ, σ) exp Normal(ln ϑ, σ) + a Normal (Gauss, Gaussian, bell curve, Laplace-Gauss, de Moivre, error, Laplaces second law of error, law of error) distribution [7, ]: { } Normal(x µ, σ) = exp (x µ) πσ σ (43) = lim Amoroso(x µ σ α, σ/ α, α, ) α = lim LogGamma(x µ σ α ln α, σ α, α) α With µ = 0 and σ = / h we obtain the error function distribution, and with µ = 0 and σ = we obtain the standard normal (Φ, z, unit normal) distribution. In the limit that σ we obtain an unbounded uniform (flat distribution, and in the limit σ 0 we obtain a delta (degenerate) distribution. The normal distribution is a limit of the Amoroso [] and log-gamma distributions [39]. For Amoroso, make the requisite substitutions, Amoroso(x µ σ α, σ/ α, α, ) { exp α x µ ( + (α ) ln + )} x µ σ α σ and expand the logarithm as ln( + x) = x x + O(x3 ). Power-law (Pearson type XI, fractal) distribution [38]: PowerLaw(x p) (x a) p (44) = lim Amoroso(a,, α, ( p)/α) α Improper (unnormalizable) power law distributions are obtained as a limit of the gamma distribution family. If p = 0 we obtain the half-uniform distribution over the positive numbers; if p = we obtain Jeffreys distribution [], used as an uninformative prior in Bayesian probability [0]. Acknowledgments: I am grateful to David Sivak, Edward E. Ayoub and Francis J. O Brien for spotting various typos and errors, and, as always, to Avery Brooks for many insightful observations. In curating this collection of distributions, I have benefited mightily from Johnson, Kotz, and Balakrishnan s monumental compendiums [, 3], Eric Weisstein s MathWorld, and the myriad pseudo-anonymous contributers to Wikipedia.

22 Appendix A: Index of distributions / generalized-x The only consistent meaning is that distribution X is a special case of the distribution generalized-x. In practice, often means add a shape parameter. standard-x The distribution X with the location parameter set to 0, scale to, and often the Weibull shape parameter β to. Not to be confused with standardized which generally indicates zero mean and unit variance. (or translated) A distribution with an additional location parame- shifted-x ter. scaled-x (or scale-x) A distribution with an additional scale parameter. inverse-x (Occasionally inverted-x, reciprocal-x, or negative-x) Generally labels the transformed distribution with x x, or more generally the distribution with the Weibull shape parameter inverted, β β. An exception is the inverse normal distribution []. log-x If x follows distribution X then either y = ln x (e.g. log-normal) or y = e x (e.g. log-gamma) follows log-x. This ambiguity arrises because although the second convention is more logical, the log-normal convention has historical precedence. reversed-x (Occasionally negative-x) The scale is negated. X of the Nth kind See X type N. Distribution Synonym or Equation χ chi χ chi-square Γ gamma Λ log-normal Φ standard normal antilog-normal log-normal Amaroso () bell curve normal BHP (4) Bramwell-Holdsworth-Pinton BHP chi (4) chi-square () Coale-McNeil generalized log-gamma Cobb-Douglas log-normal de Moivre normal

23 / 3 degenerate delta delta See normal (43) doubly exponential Gumbel double exponential Gumbel or Laplace Erlang See gamma (5) error normal error function see normal (43) exponential (8) extreme value gumbel extreme value type N Fisher-Tippett type N Fisher-Tippett (9) Fisher-Tippett type I Gumbel Fisher-Tippett type II Fréchet Fisher-Tippett type III Weibull Fisher-Tippett-Gumbel Gumbel fractal power law flat uniform Fréchet (33) FTG Fisher-Tippett-Gumbel Galton log-normal Galton-McAlister log-normal gamma (5) Gaussian normal Gauss normal generalized gamma Stacy or Amoroso generalized inverse gamma See Stacy () generalized Gumbel (38) generalized extreme value Fisher-Tippett generalized Fisher-Tippett (8) generalized Fréchet (3) generalized inverse gamma generalized gamma generalized normal nakagami generalized Rayleigh scaled chi generalized semi-normal Stacy generalized Weibull (30) GEV generalized extreme value Gibrat standard log-normal Gumbel (39) Gumbel-Fisher-Tippett Gumbel Gumbel type N Fisher-Tippett type N half-normal () half-uniform See power law (44) hydrograph Stacy hyper gamma Stacy inverse chi (6) inverse chi-square (4)

24 / 4 inverse exponential () inverse gamma (0) inverse Rayleigh (7) inverse Weibull Fréchet Jeffreys See power law (44) Laplaces second law of error normal Laplace-Gauss normal law of error normal Leonard hydrograph Stacy Lévy () log-chi-square (37) log-gamma (34) log-normal (4) log-normal, two parameter See log-normal (4) log-weibull Gumbel logarithmic-normal log-normal logarithmico-normal log-normal Maxwell (7) Maxwell-Boltzmann Maxwell Maxwell speed Maxwell m-erlang Erlang Nakagami (0) Nakagami-m Nakagami negative exponential exponential normal (43) Nukiyama-Tanasawa generalized gamma one-sided normal half-normal Pearson type III (7) Pearson type V (9) Pearson type X exponential Pearson type XI power law positive definite normal half-normal power law (44) pseudo-weibull (4) Rayleigh (6) Rosin-Rammler Weibull Rosin-Rammler-Weibull Weibull scaled chi (5) scaled chi-square (3) scaled inverse chi (5) scaled inverse chi-square (3) semi-normal half-normal shifted exponential (9) Stacy () Stacy-Mihram Amoroso standard Amoroso standard gamma

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