The Amoroso Distribution

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Tch. Not 003v (007-08-4) Th Amoroso Distribution Gavin E. Crooks gcrooks@lbl.gov Th Amoroso distribution, is a continuous, univariat, unimodal probability distribution with a smiinfinit rang. A surprisingly larg mnagri of intrsting, univariat probability distributions ar spcial cass or limiting forms of th Amoroso distribution. Amoroso(x ν,, α, ) = ( ) { α ( ) } x ν x ν Γ(α) xp ν,, + α > 0 x ν ( > 0) x ν ( < 0) (a) This distribution has four ral paramtrs; a location paramtr ν, a scal paramtr, and two shap paramtrs α and. Anothr usful paramtrization is Amoroso (x µ,, α, λ) ( = αα + λ x µ Γ(α) = Amoroso(µ λ,, α, /λ) λαλ (b) ) α { λ ( xp α + λ x µ In th limit that λ 0, th rang bcoms x [, + ] and Amoroso (x µ,, α, 0) { ( ) ( )} = αα x µ x µ Γ(α) xp α α xp (c) (Rcall that lim a 0 ( + ax) /a = x ) W will dfin th standard Amoroso distribution as StdAmoroso(x) = x x = Amoroso(x 0,,, ) = Amoroso (x,,, ) (d) Stting to yilds Parson s typ V (March) distribution 3,4 ParsonV(x µ,, α) = ( ) α+ Γ(α) x ν =Amoroso(x ν,, α, ) () If w st th shap paramtr to unity w obtain Parson s typ III (Vinci) distribution 5 7. ParsonIII(x ν,, α) = ( ) α x ν ( ) Γ(α) (3) =Amoroso(x ν,, α, ) ) λ } 0.35 0.3 0.5 0. 0.5 0. 0.05 4 6 8 () Amoroso ν α µ α λ () Parson typ V... -... - (3) Parson typ III...... (4) Nakagami...... (6) gnralizd Frécht.. n <0.. n <0 (8) gnralizd Gumbl.. n 0 (5) gnralizd Wibull.. n >0.. n >0 (7) gnralizd xtrm valu...... (4) Frécht.. <0.. <0 (33) gnralizd log gamma... 0 (9) Gumbl.. 0 (3) BHP.. π 0 (5) Wibull.. >0.. >0 (9) shiftd xponntial.... (3) log gamma x.. 0 (5) gnralizd gamma 0... (9) scald invrs-chi 0.. - (6) invrs gamma 0.. - (34) Jffrys 0.. 0 (6) gamma 0.. () scald chi 0.. (0) strtchd xponntial 0.. () Lévy 0. - (3) half normal 0. () invrs Rayligh 0. - (0) invrs xponntial 0. - (8) xponntial 0. (4) Rayligh 0. (5) Maxwll 0. 3 (6) Win 0. 4 (7) invrs chi-squar 0. - (8) invrs chi 0. - () chi 0. (7) chi-squar 0. (8) standard xponntial 0 (30) standard Gumbl 0 0 (d) standard Amoroso 0

With = w obtain th Nakagami (gnralizd normal) distribution. Nakagami(x ν,, k/, ) (4) ( ) { k ( ) } x ν x ν = xp Γ(k/) If w drop th location paramtr from Amoroso, thn w obtain th gnralizd gamma (hypr gamma, gnralizd Wibull) distribution, th parnt of th gamma family of distributions 8,9. GnGamma(x, α, ) = ( x ) α ( x ) (5) Γ(α) x > 0, > 0 =Amoroso(x 0,, α, ) If th is ngativ thn th distribution is gnralizd invrs gamma. Not surprisingly th gamma (scald-chi-squar) distribution 5,7 is a spcial cas of th gnralizd gamma, whr th scond shap paramtr is st to unity. Gamma(x, α) = Γ(α) ( x ) α x/ = ParsonIII(x 0,, α) = GnGamma(x, α, ) = Amoroso(x 0,, α, ) Instancs of th gamma distribution oftn appar in statistical physics. For xampl th Win (Vinna) distribution Win(x T ) = Gamma(x T, 4) (An approximation to th rlativ intnsity of black body radiations as a function of th frquncy). Th Erlang distribution is a gamma distribution with intgr α. Not that w obtain Amoroso by adding to th gamma distribution both a location (as in Parson typ III) and an additional shap paramtr (as in th gnralizd gamma). Important spcial cass of th gamma distribution includ th chi-squar (χ, chi squard) distribution ( x ) k/ ChiSqr(x k) = x/ (7) Γ(k/) = Gamma(x, k/) = GnGamma(x, k/, ) = Amoroso(x 0,, k/, ) and th xponntial (Parson typ X) distribution (6) Exp(x ) = x (8) = Gamma(x, ) = Amoroso(x 0,,, ) W can also obtain a shiftd xponntial distribution as a spcial cas of th Parson typ III distribution ShiftExp(x ν, ) = (9) = ParsonIII(x ν,, ) = Amoroso(x ν,,, ) Strtchd xponntial 0 StrtchdExp = ( x ) ( x ) (0) = Amoroso(x 0,,, ) Additional spcial cass of th gnralizd gamma distribution includ th chi (χ) distribution ( ) k x Chi(x k) = Γ(k/) x / = GnGamma(x, k/, ) = Amoroso(x 0,, k/, ) and scald-chi (gnralizd Rayligh) distribution. ( ) k x ScaldChi(x s, k) = Γ(k/) x s s s = GnGamma(x s, k/, ) = Amoroso(x 0, s, k/, ) () () Spcial cass of th scald-chi distribution includ th half-normal (smi-normal, positiv dfinit normal ) distribution, HalfNormal(x s) = th Rayligh distribution x s (3) πs = ScaldChi(x s, ) = GnGamma(x s, /, ) = Amoroso(x 0, s, /, ) Rayligh(x s) = x x s s (4) = ScaldChi(x s, ) = GnGamma(x s,, ) = Amoroso(x 0, s,, ) and th Maxwll (Maxwll-Boltzmann) distribution Maxwll(x s) = πs 3 x x s (5) = ScaldChi(x s, 3) = GnGamma(x s, 3/, ) = Amoroso(x 0, s, 3/, ) With ngativ shap paramtrs, th gnralizd gamma gnrats various invrs distributions, including th invrs gamma (scald invrs chi-squar ) distribu-

3 tion, InvGamma(x, α) = Γ(α) th invrs-chi-squar distribution, InvChiSqr(x k) = th invrs-chi distribution, ( ) α+ /x (6) x = GnGamma(x, α, ) = ParsonV(x 0,, α) = Amoroso(x 0,, α, ) ( ) k + x (7) Γ(k/) x = InvGamma(x /, k/) = GnGamma(x /, k/, ) = ParsonV(x 0, /, k/) = Amoroso(x 0, /, k/, ) InvChi(x k) = ( ) k+ x Γ(k/) (8) x scald invrs-chi distribution, = GnGamma(x /, k/, ) = Amoroso(x 0, /, k/, ) ScaldInvChi(x s, k) = s invrs xponntial, Γ(k/) ( ) k+ s x s x (9) = GnGamma(x / s, k/, ) = Amoroso(x 0, / s, k/, ) InvExp(x ) = x /x (0) and invrs Rayligh. = InvGamma(x, ) = GnGamma(x,, ) = Amoroso(x 0,,, ) InvRayligh(x ) = ( ) 3 8s s x x () = GnGamma(x / s,, ) = Amoroso(x 0, / s,, ) Th Lévy distribution (Van dr Waals profil) is a spcial cas of th invrs gamma distribution. Th Lévy distribution is notabl for bing stabl; a linar combination of idntically distributd Lévy distributions is again a Lévy distribution. Th othr stabl distributions with analytic forms ar th normal (which w ncountr blow) and th Cauchy distribution, which is not a mmbr of th Amoroso family. c c/x Lévy(x c) = () π x 3/ = InvGamma(x c/, /) = GnGamma(x c/, /, ) = ParsonV(x 0, c/, /) = Amoroso(x 0, c/, /, ) Th Wibull (xtrm valu typ III, Fishr-Tipptt typ III, Gumbl typ III) distribution,3 occurs with th shap paramtr α =. This is th limiting distribution of th minimum of a larg numbr idntically distributd random variabls that ar at last ν. (Maximum if is ngativ.) Wibull(x ν,, ) = ( x ν = Amoroso(x ν,,, ) ) ( ) (3) Spcial cass of th Wibull distribution includ th xponntial ( = ) and Rayligh ( = ) distributions, and th standard Wibull (ν = 0). Th Frécht (xtrm valu typ II, Fishr-Tipptt typ II, Gumbl typ II, invrs Wibull) distribution is th limiting distribution of th largst of a larg numbr idntically distributd random variabls whos momnts ar not all finit and ar boundd from blow by ν. (If th shap paramtr is ngativ thn minimum rathr than maxima.) ( ) Frécht(x ν,, ) = + x ν ( ) = Amoroso(x ν,,, ) (4) Spcial cass of th Frécht distribution includ th invrs xponntial ( = ) and invrs Rayligh ( = ) and th standard Frécht (ν = 0) distribution. Instad of asking for th minimum or maximum of a larg numbr of random variabls, w instad ask for th nth largst w obtain th gnralizd Wibull distribution GnWibull(x ν,, n, ) = ( ) n x ν and th gnralizd Frécht distribution. GnFrécht(x ν,, n, ) = ( )n = Amoroso(x ν,, n, ) (5) ( ) n + x ν ( )n = Amoroso(x ν,, n, ) (6) Th gnralizd xtrm valu (GEV, von Miss- Jnkinson) distribution is th suprclass of typ I, II and

4 III xtrm valu distributions. GnExtrmValu(x µ,, λ) (7) = ( + λ x µ ) { λ ( xp + λ x µ ) } λ = Amoroso (x µ,,, λ) = Amoroso(µ λ,,, /λ) λαλ Th gnralizd Gumbl (gnralizd log-gamma) distribution is th limiting distribution of th nth largst valu of a larg numbr of unboundd idntically distributd random variabls. GnGumbl(x µ,, n) (8) = ( ) ( )} µ x µ x {n xp n xp = Amoroso (x µ,, n, 0) If w limit n = thn w obtain th Gumbl (Fishr- Tipptt (typ I), Fishr-Tipptt-Gumbl, FTG, Gumbl- Fishr-Tipptt, log-wibull, xtrm valu (typ I), doubly xponntial) distribution Gumbl(x µ, ) (9) = {( ) ( )} x µ x µ xp xp = Amoroso (x µ,,, 0) With ngativ scal < 0, this is an xtrm valu distribution of maximum, with > 0 an xtrm valu distribution of minima. (Not that oftn th Gumbl is dfind with th ngativ of th scal usd hr.) A Gomprtz distribution is a truncatd Gumbl. Th standard Gumbl (Gumbl) distribution is StdGumbl(x) = xp {x x } (30) =Amoroso (x 0,,, 0) Anothr spcial cas of th gnralizd Gumbl is th BHP (Bramwll-Holdsworth-Pinton) distribution 4,5 BHP(x µ, ) (3) = { ( ) π x µ xp π ( )} x µ xp = Amoroso (x µ,, π, 0) Log-gamma LogGamma(x, α) = { ( x ) ( x )} Γ(α) xp α xp =Amoroso (x ln α,, α, 0) (3) Gnralizd Log-Gamma(Coal-McNil 6,7 ) GnLogGamma(x µ,, α) (33) = { ( ) ( )} x µ x µ Γ(α) xp α xp = Amoroso (x µ + ln α,, α, 0) If w lt = 0 thn w obtain Jffrys distribution 8, an impropr (unnormalizabl) distribution widly usd as an uninformativ prior in Baysian probability 9. Jffrys(x) x = Amoroso(0,, α, 0)) (34) If and α ar finit, thir xact valus ar irrlvant. If w tak th limit α but kp th product α = p constant thn w can obtain a varity of impropr powrlaw (Parson typ XI 0, fractal) distributions. PowrLaw(x p) x p (35) = lim Amoroso(0,, α, ( p)/α) α If p = 0 w obtain th half-uniform distribution ovr th positiv numbrs. Th normal (Gauss, Gaussian, bll curv) distribution can b obtaind in svral limits. For xampl, Normal(x µ, ) (36) { } = xp (x µ) π = lim α Amoroso (x µ, / α, α, 0) In th limit that w obtain an unboundd uniform distribution, and in th limit 0 w obtain a dlta function distribution. Proprtis ( ) n x ν E[ ] = Γ(α + n ) Γ(α) man = ν + Γ(α + ) Γ(α) [ Γ(α + varianc = ) Γ(α + ) Γ(α) Γ(α) ) Entropy = log Γ(α) + α + ( α ] ψ(α) (37) (38) (39) (40)

5 Indx of distributions Distribution Equation χ.............................................. S chi χ...................................... S chi-squar Γ.......................................... S gamma Amaroso...........................................(a) bll curv.................................. S normal BHP...............................................(3) Bramwll-Holdsworth-Pinton................. S BHP chi.................................................() chi-squar...........................................(7) chi-squard.............................. S chi-squar Coal-McNil...............S gnralizd log-gamma dlta..............................................(36) doubly xponntial........................ S Gumbl Erlang......................................S gamma xponntial........................................ (8) xtrm valu typ N....... S Fishr-Tipptt typ N Fishr-Tipptt typ I...................... S Gumbl Fishr-Tipptt typ II...................... S Frécht Fishr-Tipptt typ III..................... S Wibull Fishr-Tipptt-Gumbl.................... S Gumbl fractal.................................. S powr law flat!.......................................s uniform Frécht........................................... (4) FTG....................... S Fishr-Tipptt-Gumbl gamma............................................. (6) Gaussian...................................S normal Gauss......................................S normal gnral gnralizd gamma............... S Amoroso gnralizd gamma................................. (5) gnralizd log-gamma............................ (33) gnralizd Gumbl............................... (8) gnralizd xtrm valu......................... (7) gnralizd Frécht................................(6) gnralizd invrs gamma..... S gnralizd gamma gnralizd normal...................... S Nakagami gnralizd Rayligh.....................S scald-chi gnralizd Wibull................................ (5) GEV.................... S gnralizd xtrm valu Gomprtz................................. S Gumbl Gumbl........................................... (9) Gumbl-Fishr-Tipptt.................... S Gumbl Gumbl typ N............. S Fishr-Tipptt typ N half-normal....................................... (3) half-gaussian..........................s half-normal half-uniform...................................... (35) hypr gamma.................. S gnralizd gamma invrs chi........................................ (8) invrs chi-squar................................. (7) invrs xponntial................................(0) invrs gamma.................................... (6) invrs Rayligh.................................. () invrs Wibull............................ S Frécht Jffrys........................................... (34) Lévy.............................................. () log-gamma........................................ (3) log-wibull................................s Gumbl March............................. S Parson typ V Nakagami.......................................... (4) normal............................................(36) Parson typ III................................... (3) Parson typ V.................................... () Parson typ X........................ S xponntial Parson typ XI........................ S powr law positiv dfinit normal................ S half-normal powr law.........................................(35) Rayligh.......................................... (4) Maxwll.......................................... (5) Maxwll-Boltzmann...................... S Maxwll shiftd xponntial................................. (9) scald chi.......................................... () scald chi-squar........................... S gamma scald invrs chi..................................(9) scald invrs chi-squar............s invrs gamma smi-normal........................... S half-normal standard Amoroso................................ (d) standard Gumbl................................. (30) strtchd xponntial............................. (0) uniform........................................... (36) Van dr Waals............................... S Lévy Vinna.......................................S Win Vinci............................. S Parson Typ V von Miss-Jnkinson......... gnralizd xtrm valu Wibull........................................... (3) Win....................................... S gamma Amoroso, Annali di Mathmatica, 3 (95). N. L. Johnson, S. Kotz, and N. Balakrishnan, Continuous Univariat Distributions, vol. (Wily, Nw York, 994), nd d. 3 K. Parson, Philos. Trans. R. Soc. A (894). 4 K. Parson, Philos. Trans. R. Soc. A 97, 443 (90). 5 M. Abramowitz and I. A. Stgun, Handbook of Mathmatical Functions with Formulas, Graphs, and Mathmatical Tabls (Dovr, Nw York, 964). 6 K. Parson, Philos. Trans. R. Soc. A (893). 7 K. Parson, Philos. Trans. R. Soc. A 86, 343 (895). 8 E. W. Stacy, Ann. Math. Stat 33, 87 (96). 9 A. Dadpay, E. S. Soofi, and R. Soyr, J. Economtrics 38, 568 (007). 0 J. Lahrrèr and D. Sorntt, Eur. Phys. J. B, 55 (998). A. Glman, J. B. Carlin, H. S. Strn, and D. B. Rubin, Baysian Data Analysis (Chapman & Hall/CRC, Nw York, 004), nd d.

6 N. L. Johnson, S. Kotz, and N. Balakrishnan, Continuous Univariat Distributions, vol. (Wily, Nw York, 995), nd d. 3 W. Wibull, J. Appl. Mch.-Trans. ASME 8, 93 (95). 4 S. T. Bramwll, P. C. W. Holdsworth, and J.-F. Pinton, Natur 396 (998). 5 S. T. Bramwll, K. Christnsn, J.-Y. Fortin, P. C. W. Holdsworth, H. J. Jnsn, S. Lis, J. M. Lópz, M. Nicodmi, J.-F. Pinton, and M. Sllitto, Phys. Rv. Ltt. 84, 3744 (000). 6 A. Coal and D. R. McNil, J. Amr. Statist. Assoc. 67, 743 (97). 7 R. Kanko, Dm. Rs. 9, 3 (003). 8 H. Jffrys, Thory of probability (Oxford Univrsity Prss, 948), nd d. 9 E. T. Jayns, Probability Thory: Th Logic of Scinc (Cambridg Univrsity Prss, Cambridg, 003). 0 K. Parson, Philos. Trans. R. Soc. A 6, 49 (96).

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