An Exceptional Generalization of the Poisson Distribution

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1 Open Journal of Saisics, 1,, hp://dx.doi.org/1.436/ojs Published Online July 1 (hp:// An Excepional Generalizaion of he Poisson Disribuion Per-Erik Hagmark Deparmen of Mechanics and Design, Tampere Universiy of Technology, Tampere, Finland per-erik.hagmark@u.fi Received May 4, 1; revised June 1, 1; acceped June 3, 1 ABSTRACT A new wo-parameer coun disribuion is derived saring wih probabilisic argumens around he gamma funcion and he digamma funcion. This model is a generalizaion of he Poisson model wih a noeworhy assormen of qualiies. For example, he mean is he main model parameer; any possible non-rivial variance or zero probabiliy can be aained by changing he oher model parameer; and all disribuions are visually naural-shaped. Thus, exac modeling o any degree of over/under-dispersion or zero-inflaion/deflaion is possible. Keywords: Coun Daa; Gamma Funcion; Poisson Generalizaion; Discreizaion; Modeling; Over/Under-Dispersion; Zero-Inflaion/Deflaion 1. Inroducion and he Main Resul In coun daa modeling he Poisson disribuion is usually he firs opion, bu real daa can indicae a variey of discrepancies. These can be genuine feaures or secondary consequences of e.g. censoring, clusering, approximaions or correlaions. Specifically, he Poisson model has no dispersion flexibiliy because he mean deermines he variance and he zero probabiliy, σ = μ, p = e μ, while he real daa can display over or under- dispersion, σ μ, or zero-inflaion or deflaion, p e μ [1]. Such siuaions are usually handled e.g. by randomizing he Poisson mean, by mixures, by adding a new parameer, by reweighing he Poisson poin probabiliies, or via generalizing he exponenial incremens in he homogeneous Poisson process [-5]. Our approach will be differen. We recall an elemenary fac. The mean-deviaion pair (μ, σ) of a non-binary coun variable (non-negaive ineger-valued random variable) always saisfies he inequaliy 1, (1) where [μ] is he larges ineger no exceeding μ. Thus, we will say ha a coun model (parameerized coun variable) has full dispersion flexibiliy if every posiive soluion (μ, σ) of he inequaliy (1) is he mean-deviaion pair for some parameer values. In [6] we called for a mahemaically unified coun model N(μ, β) wih wo independen parameers, µ >, β >, and he following properies: 1) Comforable parameerizaion: E(N(μ, β)) = μ, for all μ and β. ) Generalizaion of he Poisson model: For β = 1, n Pr N μ,1 n e n!, n =, 1,. 3) Full dispersion flexibiliy: If he numbers μ > and σ > saisfy inequaliy (1), hen here is a β such ha Var N,. The soluion o be presened in his paper obeys he following cumulaive probabiliies: Pr N μ, β n n n 1 ng, ng, 1 () n1 n1 n1 G, n1 g, 1, where g(, x) and G(, x) are he one-parameer gamma probabiliy and cumulaive disribuion funcions, respecively, wih parameer x and variable (Secion ). We begin wih he derivaion of fundamenal inequaliies in Secion. These inequaliies lead o a cumulaive disribuion H(x, μ), where he parameer μ > is he mean. Then he inserion of a new independen parameer β > provides an exended cumulaive disribuion H(x/β, μ/β) and he relaed non-negaive wo-parameer random variable X(μ, β), where μ is sill he mean. Now he proclaimed coun model N(μ, β) is defined as a mean-preserving discreizaion of X(μ, β), and he above properies 1), ), 3) are proved. Thereafer he mos immediae applicaions are given; namely, exac modeling of over/ under-dispersion or zero-inflaion/deflaion o any possible degree. In he las secion, we propose moives for furher research, and we compare N(μ, β) wih well-esablished Poisson generalizaions. Copyrigh 1 SciRes.

2 314 P.-E. HAGMARK. Derivaion of Two Inequaliies We sar wih noaion: Gamma funcion Г(x) as Euler s second inegral, digamma funcion Ψ(x), some relaed funcions and immediae inerrelaions; x x1 : e d, x, x: x x, x 1, : e g x x,, G, x: g s, x d s, ax (, ): gx, gx, ln x, x bx, : ax, gx, ln x x, x Ax, : Gx, asx, )d s, x x B, x: A, x b s x, d s. There is a nice probabilisic perspecive on he gamma funcion: If he random variable T has a gamma densiy g(, x), hen E(ln(T)) = Ψ(x) and Var(ln(T)) = dψ(x)/dx [7]. In erms of our noaion above, hese simple observaions can be wrien in he form lim Ax,, lim Bx,, x. (3) Addiional work leads o a sronger resul, Ax, d 1, Bx, d, x. (4) Namely, inegraion by pars, he funcional equaions x 1 xx, gx, xgx, 1, formula (3), and l Hospial s rule allow us o wrie Ax, d Ax gx, 1ln x x1 x x 1, x d lim Bx, gx, ln x x g, x 1 ln x x x1x lim (, ), ln x d x g x B, x d x x x d d. Nex we derive wo fundamenal inequaliies. For every fixed x >, he funcion a(, x) has exacly one roo x e, and i is increasing here. This and (3, lef side) imply Ax,,, x. 1 Ax, d, x,. (5) Now, aking ino accoun (5) and (4, lef side), we obain he firs inequaliy, 1 (6) Furher, for every fixed x >, he funcion b(, x) has x x exacly wo roos, e x e B, x d, x,. x, and i is decreasing a and increasing a 1. From his one can conclude ha B(, x) has, for every x >, a posiive local maximum a and, because of (3, righ side), a negaive local minimum a 1. Considering (4, righ side) oo, we finally arrive a he second inequaliy 3. A Mean-Preserving Discreizaion We will also need a cerain discreizaion procedure: If X is a non-negaive random variable wih cumulaive disribuion F(x), he discreizaion of X is a coun variable N wih cumulaive probabiliies equal o he mean F(x) on he inerval (n, n + 1), i.e. n1 Pr N n : F x d x, n,1, n We shorly quoe he basic properies from [6]: The mean and he variance of N exis (are finie) if and only if he mean and he variance of X exis, and in ha case N X X N X X (7) (8) E E, (9) Var Var Var min E, A Generalizaion of he Poisson Model (1) In our consrucion of a new generalizaion of he Poisson model, he following one-parameer funcion will be he cenral ingredien: H x, : 1 G, xd. (11) x Recalling (5) and he noaion A(, x) = G(, x)/ x from Secion, we derive 1 Hx, d x A, xdx d. (1) In (1) we firs changed he inegraion order (as he inegrand is posiive) and hen employed he limis G x G, : lim x, 1, (13a) G x G, : lim, (13b) x. Copyrigh 1 SciRes.

3 P.-E. HAGMARK 315 The limis (13) follow from Chebyshev s inequaliy and he simple fac ha he parameer x of he one-parameer gamma densiy g(, x) equals he mean and he variance. By employing he inequaliies (6) and (7), we have < H(x, μ) < 1 and H(x, μ)/ x >. Hence, H(x, μ) is a cumulaive probabiliy disribuion wih mean μ (1) and zero probabiliy H, : lim x Hx,. We proceed by adding an independen parameer β >, so defining a wo-parameer cumulaive disribuion, x Fx,, : H,, x. (14) Now, le X(μ, β) be he non-negaive random variable deermined by F(x, μ, β), and le N(μ, β) be he discreizaion of X(μ, β), according o Secion 3. We form an inegral funcion of (14) and ge he cumulaive probabiliies of N(μ, β) using (8): I x,, : F x,, dx x x Gs, ds x x G, d,,, : Pr, ) In In P n N n 1,,,, (15) (16) n n1 1 G, G, d The pair X(μ, β) and N(μ, β) is illusraed in Figure 1. Proof of Properies 1) and ), Secion 1. By considering (9, 1, 14) one can see ha he mean does no change during he process from H(x, μ) o N(μ, β): x E N μβ, E X μβ, 1 H, dx β β (17) E X,1, proving Propery 1). Nex, we fix β = 1 in (16) and employ he ideniies G(, x) G(, x + 1) = g(, x + 1) and G(, ) = 1 (13a). Now Pr{N(μ, 1) n} = 1 G(μ, n + 1), so he poin probabiliies are Pr{N(μ, 1) = n} = G(μ, n) n G(μ, n + 1) = e n!, n =, 1,. This means ha he sub-model N(μ, 1) is he Poisson model, so Propery ) holds rue (see case β = 1 in Figure 1). 5. Full Dispersion Flexibiliy Propery 3), Secion 1, remains o be proved. Given any posiive pair (μ, σ) saisfying 1, we have o prove ha here is a β > such ha Var(N(μ, β)) = σ. Figure is an illusraion. Firs, one obains an upper bound for he variance of X(μ, β) by employing Properies 1) and ), (1, lef side) and rouines: Var X, x 1 H x, dx E X,1 Var X,1 N Var,1. (18) Then (18) and (1, righ side) imply Var(N(μ, β)) <. Afer noing ha Var(N(μ, β)) is a coninuous funcion of β (for fixed μ) and recalling inequaliy (1), i is enough o prove he following limis: lim Var N, 1, (19) lim Var N, () Figure 1. Cumulaive disribuions of X(μ, β) and N(μ, β), for μ = 3. and β = 1,.6, 4,.1. Copyrigh 1 SciRes.

4 316 P.-E. HAGMARK Figure. The variance Var(N(μ, β)) as a funcion of β, for μ = 3. and μ =.7. Poisson poin (β = 1, σ = μ); lower bound = μ μ 1 μ+ μ. σmin Proof of (19). From (18) i follows ha Var(X(μ, β)) ends o zero as β. This means ha X(μ, β) approaches he consan µ (in disribuion). This again means ha he discreizaion N(μ, β) approaches μ if his is an ineger, and oherwise a binary coun variable wih he values [μ] and [μ]+1; see [6]. In boh cases he limi of Var(N(μ, β)) obeys (19). Proof of (). Definiion (11) and parial inegraion yield he ideniy M x 1 H x, dx M M G, Md G x x, d d. The firs erm on he righ side vanishes when M, since MG(, M) M /Г(M). Now by changing he inegraion order in he laer erm, one obains where E X,1 x 1 H x, dx L s ds d, s x1 L s : g s, x d xe s x dx. (1) Then, by using (1) and par of (18), and changing inegraion variable, z = β, one arrives a X X,1 E, E z z Lsdsd z. z x 1 Furher, he inequaliy s 1 x1 ln s >, yields a lower bound for L(s): 1 L s g s, x dxe x s C Dln s, x 1 C dx, D dx. x (), s >, x This means ha L(s) ends o as s, and so he average of L in he inerval (, z/β) approaches as β (). Thereby, E(X(μ, β) ) grows o, so (17) and (1, lef side) complee he proof of (). 6. Compuing and Applicaions When working wih N(μ, β), he following numbers are useful: K n, n : G, d n n ng, ng, 1, n,1, (3) The laer faser version follows from parial inegraion and he ideniies G(, x) G(, x + 1) = g(, x + 1), G(, ) = 1 (13a). Noe also ha mos mahemaical sofware offers fas compuaion of G(, x). Employing (3) in (16), basic formulas can be wrien in he following form: N n Kn Kn 1 Pr, ) 1,,, E N, n1 k k k k n n n Kn 1 1,, N Kn n1 4) (5) Var,,. (6) We consider exac modeling of coun variables. (For numerical examples, see Table 1). Applicaion 1. Generally, a non-binary coun variable wih desired mean μ and variance σ exiss if and only if 1. (7) In ha case N(μ, β) always provides a soluion. Indeed, because of full dispersion flexibiliy, Propery 3), here Copyrigh 1 SciRes.

5 P.-E. HAGMARK 317 Table 1. Under/over-dispersion and zero-deflaion/inflaion. Phenomenon General range Numerical example Soluion Under-dispersion (μ [μ])(1 μ + [μ]) < σ < μ μ = 3. σ =.4 β =.753 Poisson σ = μ (equi-dispersion) μ = 3. σ = 3. β = 1 Over-dispersion μ < σ < μ = 3. σ = 4.5 β = Zero-deflaion max{,1 μ}< p < e μ μ = 3. p =.1 β =.56 Poisson p = e μ μ = 3. p = β = 1 Zero-inflaion e μ < p < 1 μ = 3. p =.15 β =.949 is a β > such ha Var(N(μ, β)) = σ (6). Applicaion. Likewise, a non-binary coun variable wih desired mean μ and zero probabiliy p exiss if and only if p max, 1 1. (8) Again N(μ, β) provides a soluion. Argumens like hose in Secion 5 would show ha here is a β > such ha Pr N, ) = p (4, n = ). Applicaion 3. Suppose here is a real non-censored random sample available of he unknown non-binary coun variable o be modeled. Le ˆ be he sample mean, ˆ he sandard variance and ˆp he zero fracion. I is easy o prove ha hese UMVU esimaes also mee (7, 8). Thus, here is a β 1 ha saisfies ˆ and a β ha saisfies ˆp (boh exacly), bu of course, usually 1. Imporance weighing provides a compromise β and an approximae soluion N ˆ,. 7. Furher Research and Discussion Addiional work is needed o enlarge he applicabiliy of N(μ, β). The compuaional behavior of he cenral formulas 3-6 should be furher explored, and ools for sochasic simulaion and saisical inference should be developed. We pu forward wo concree problems. Problem 1. Numerical experimenaion indicaes ha he numbers K n (3, n 1) increase wih β (K = μ). If his is rue, all momens (5, k ) increase wih β, so he ieraion of β in he applicaions in Secion 6 can be made faser. Problem. Find an algorihm for generaion of random variaes from N(μ, β). The alias mehod [8] can of course be used for runcaed versions, bu a ailor-made mehod would be welcome. Acually, a generaion mehod for X(μ, β) would be enough since, according o [6], his can immediaely be ransformed o he discreizaion N(μ, β). Finally, we reurn o he main qualiies of N(μ, β). As menioned, he finie mean-deviaion pair (μ, σ) of any non-binary coun variable saisfies inequaliy (1), i.e. σ > 1. Conversely, if (μ, σ) is a posiive soluion of (1), hen i is he mean-deviaion pair of a non-binary coun variable; and as we have shown, here is always an N(μ, β) wih his mean-deviaion pair. Since he mean is an original model parameer of N(μ, β), only β needs o be solved from he equaion Var(N(μ, β)) = σ. We have called his feaure full dispersion flexibiliy, because i enables exac modeling for he firs wo momens, or for mean and zero probabiliy. Full dispersion flexibiliy seems o be very rare even among well-esablished Poisson generalizaions. The generalizaion of Consul and Jain [], he negaive binomial [3], he COM-Poisson disribuion [4] and many ohers have severe shorcomings in dispersion flexibiliy, and also parly bad-shaped disribuion funcions. A posiive excepion is he General Poisson Law [5]. However, here he mean is no a model parameer, so, if a cerain pair (μ, σ) is waned, he original parameers mus be solved simulaneously from wo equaions, which boh include laborious infinie series. Also noe ha he invarians (4) and (5), he inequaliies (6) and (7), and he disribuion (11) comprise, as such, a conribuion o probabilisic reamen of he gamma funcion. REFERENCES [1] J. Casillo and M. Perez-Casany, Over-Dispersed and Under-Dispersed Poisson Generalizaions, Journal of Saisical Planning and Inference, Vol. 134, No., 5, pp doi:1.116/j.jspi [] P. C. Consul and G. C. Jain, A Generalizaion of he Poisson Disribuion, Technomerics, Vol. 15, No. 4, 1973, pp doi:1.37/ [3] N. L. Johnson, S. Koz and A. W. Kemp, Univariae Discree Disribuions, nd Ediion, John Wiley & Sons, New York, 199. [4] R. W. Conway and W. L. Maxwell, A Queuing Model wih Sae Dependen Service Raes, Journal of Indusrial Engineering, Vol. 1, 196, pp [5] G. Morla, Sur Une Généralisaion de la loi de Poisson, Compes Redus, Vol. 35, 195, pp [6] P.-E. Hagmark, On Consrucion and Simulaion of Coun Copyrigh 1 SciRes.

6 318 P.-E. HAGMARK Daa Models, Mahemaics and Compuers in Simulaion, Vol. 77, No. 1, 8, pp doi:1.116/j.macom [7] L. Gordon, A Sochasic Approach o he Gamma Funcion, The American Mahemaical Monhly, Vol. 11, No. 9, 1994, pp [8] A. J. Walker, An Efficien Mehod for Generaing Discree Random Variables wih General Disribuions, ACM Transacions on Mahemaical Sofware, Vol. 3, 1977, pp doi:1.1145/ Copyrigh 1 SciRes.