The Generalized Power Series Distributions and their Application

Transcription

1 Fereshteh Momei/ TJMCS Vol.2 No.4 (20) The Joural of Mathematics ad Computer Sciece Available olie at The Joural of Mathematics ad Computer Sciece Vol.2 No.4 (20) The Geeralized Power Series Distributios ad their Applicatio Fereshteh Momei Islamic Azad Uiversity-Behshahr Brach Received: August 200, Revised: November 200 Olie Publicatio: Jauary 20 Abstract Necessity of the use of more rich family discrete distributios is practically observable. I this paper, we study about ew family of discrete distributios, as Iflated Parameter distributios. The, we use the applicatio of these distributios for data modelig of Isurace claims ad idicate that these distributios have more appropriate approximatio the correspodig distributios. Keywords: Geeralized Power Series Distributio, zero-iflated distributio.. Itroductio The importat distributios are extesively used i coutig: Biomial, Negative biomial ad Poisso distributios. Accordig to be i appropriatig classic distributios i modelig of some frequecy data or depeded data about differet areas as Ecoomics, Fiace,Isurace, Ecological ad etc..., Bowers et.al [5], Rolsi et.al [6], Wielma [6] ad Kumer et.al [7] have emphasized the geeralized ecessity of these distributios. Classic distributios are i doubt about Isurace data modelig. Here, i additio to itroducig Iflated-Parameter Power Series distributios, we will idicate the preferece of these distributios o correspodig classic distributio by usig a real data set related to the umber of accidet Isurace claims. 69

2 Fereshteh Momei/ TJMCS Vol.2 No.4 (20) GPS distributios Several ow uivariate discrete probability distributios belog to the class of geeralized power series (GPS) distributios. The geeral from of the probability mass fuctio (pmf) of a radom variable (r.v) N of a GPS distributio with parameter θ > 0 is give by P N = = a θ g θ, S, θ > 0 () Where S is ay oempty eumerable set of oe egative itegers, a 0 ad g θ = S a θ is the ormalizig costat. Their correspodig probability geeratig fuctio (pgf) of N has the followig from P N t = g θt g t, t 0, (2) The Bi(,),NB(r,), Po(λ) ad LS() distributios belog to this class. This is illustrated i Table () below. Table () θ a() g(θ) S pmf pgf Bi, + θ 0,,, + t Po λ λ! e θ 0,,2, λ! e λ t e NB r, + θ r 0,,2, r + r t r LS l θ,2,3, l l t l Amog the elemets studied i Ris Theory is the r.v. sums N = X + X X N, S 0 = 0, called Compoud r.v. It describes the aggregate claim amout i the ris model. We assume that N, X, X 2, are mutually idepedet r.v. ad say that S N has a compoud distributio. If we suppose that the r.v. N has the GPS distributios. The the relatio (3) of the pmf is useful for derivig the compoud distributio. I Pajer (98) is show that the pmf of GPS distributios satisfies the followig recursio: p = a + b p, =,2,, a <, b 0 692

3 Fereshteh Momei/ TJMCS Vol.2 No.4 (20) I the particular cases, the costats a ad b are give by the followig expressios : X~Bi, : a =, b = + X~Po λ : a = 0, b = λ X~NB r, : a =, b = r 3. Zero-Iflated GPS distributios Gupta et al (995), Kolev et al (2000) ad Miova (2002) itroduced ad studied iflated extesios of GPS distributios. Let ξ be a arbitrary oe egative iteger-valued r.v. such that P ξ = j = p j, j = 0,,2, ad j=0 p j =. Also, Let P ξ t = E t ξ be its pgf. A extra proportio of zeros ρ 0, is added to the proportio of zeros from the distributio of the r.v. ξ, while decreasig the remaig proportios i a appropriate way. The zero-iflated modificatio η of ξ is defied by P η=j = ρ+ -ρ p 0, j=0 -ρ p j, j=, 2 (3) The the pgf η is P η t =ρ+ -ρ P ξ t. Ifρ =, the the correspodig zero-iflated distributios is degeerated at zero adρ = 0, there is o iflatio i.e., P η t =P ξ t. Geerally, the iflatio parameter ρ may also tae egative values, provided that P η=0 0, i.e. ρ -p 0 ad therefore max, ρ 0. This case correspods -p 0 to the opposite Pheomea i.e excludes a proportio of zeros from the basic discrete distributio if ecessary. I Table (2) below zero-iflated couterparts of ow GPS (ZIGPS) distributios with their correspodig pmf ad pgf are summarized. Table (2) P η = 0 P η = 0, S 0 pgf ZIBi,, ρ ρ + ρ ρ j j ρ + ρ + t ZIPo λ, ρ ρ + ρ e λ ρ λj j! e λ λ t ρ + ρ e ZINB r,, ρ ρ + ρ ρ j ρ + ρ t r 693

4 Fereshteh Momei/ TJMCS Vol.2 No.4 (20) ZILS, ρ ρ ρ j j l ρ + ρ l t l 4. Iflated-Parameter GPS distributios Let W, W 2, be a sequece of ZIBi (,, ρ) with the iflated parameter max, ρ. Also, let r.v. V be the umber of trials that we eed to achieve the first observed success i the sequece W, W 2, of idepedet biary variables. The pmf of the r.v. V is give by P V = K = ρ + ρ, =,2,3, Now, defie the r.v. S by the followig relatios P S = =, = 0 ρ + ρ ρ, =,2,3, (4) Where 0, ad ρ max,,. The S is called a Iflated-Parameter Geometric distributed r.v. ad is deoted by ~IGe, ρ. The pgf of S~IGe, ρ is give by P S t = tρ t +ρ (5) Which is actually the pgf of S = X + X X N where X, X 2,., X N are Ge ρ distributed i.i.d r.v..'s ad are idepedet of N~Ge. If N is a r.v. distributed accordig to the Bi,, NB r,, Po λ ad LS distributio, the S = X + X X N is distributed from the IBi,, ρ, INB r,, ρ, IPo λ, ρ ad ILS, ρ, respectively, where X i s are Ge ρ distributed i.i.d r.v.'s ad are idepedet of N. The r.v. S belogs to the family of Iflated-Parameter GPS distributios, if its pmf ca be represeted as P S = = g θ, 2, c θ ρ ρ (7) With = 0,,2,3, ad ρ 0,, θ > 0 ad the summatio is o the set of all oegative itegers, 2, such that =. If X~ILS, ρ, the its realizatios begi from ad the summatio is over the oegative itegers such that = +. Also, the pgf of S has the followig from 694

5 Fereshteh Momei/ TJMCS Vol.2 No.4 (20) P S t = g θp X t g θ, t 0, (8) Where P X t = t ρ tρ, t 0, is the pgf Ge ρ. The scod from of pmf, which ca be foud by Taylor expasio of the pgf, is give by The correspodig θ, g θ, c, a, pgf P S, H, P S = 0, mea ad variace for specified distributios are give i Table (3). The pmf of Iflate arameter Power Series distributios satisfies the followig recursio: p = a + b p + 2 c p 2, =,2,, p = 0 the particular cases, the costats a, b ad c are give by the followig expressios : S~IBi,, ρ : a = 2ρ ρ, b = + ρ 2ρ, c = ρ ρ ρ S~IPo λ, ρ : a = 2ρ, b = λ ρ 2ρ, c = ρ 2 S~INB r,, ρ : a = ρ + 2ρ, b = r ρ 2ρ, c = ρ ρ 5. Applicatio I this chapter, we approximate the classic poisso distributio ad Iflated-Parameter poisso distributio to a real data group that is related to the umber of accidet Isurace claims ad it has prepared o the basis of 6760 policies (umber of policyholders) i Mazadara provice, we idicate that Iflated-Parameter poisso distributio is strogly appropriate. The Table of frequecy data is followig: o Ad the, X = 2.02, S 2 = Fit of Classic Poisso distributio Assumed that p is correspodig probability of class t, ad e is expected values of class t. I this case, with estimatig parameter λ = X = 2.02, statistics goodess of fit χ 2 for poisso distributio is calculated o Table (3). The, the assumptio of data poisso i level 0.05 ad eve i sigificace level is strogly rejected, because the observed value χ 2 = 40.4 is bigger tha critical poit χ ,3 =

6 Fereshteh Momei/ TJMCS Vol.2 No.4 (20) Fit of Iflated-Parameter poisso distributeo Accordig to the mea ad variace distributio IPo λ, ρ i Table (2), we obtai the followig momet estimates for the parameters, ρ : ρ = S2 X S 2 +X ad λ = 2 X 2 S 2 + X The, we have λ = 0.959,ρ = 0.03, ad therefore, the value calculatio Table of goodess of fit statistic χ 2, i this maer will obtaied as Table (5). The, agaist before maer, sice χ 2 =.407, χ 2 = 5.99, Null hypothesis based o fit of 0.05,2 Iflated-Parameter poisso distributio to data is strogly accepted. We remid that here the positiveess of estimate ρ = 0.03 is idicated that the observace of zero are more tha this that it is forcasted by classic poisso distributio. Table (3) IBi,, ρ IPo λ, ρ INB r,, ρ ILS, ρ θ λ g(θ) + θ e θ θ r l θ c() 0,, 2,! 2! r, 2,, r !! 2! a(i) i i! + i i i pgf t tρ λ t e tρ tρ t + ρ r l l ρt t + ρ H mi, P(S=0) e λ r 0 Mea Variace ρ + ρ ρ 2 λ ρ λ + ρ ρ 2 r ρ r + ρ 2 ρ 2 ρ l l + ρ + 2 ρ 2 l 2 696

7 Fereshteh Momei/ TJMCS Vol.2 No.4 (20) Table (4) K sum o e e o / χ 2 = 40.4 e Table (5) K sum o e e o / χ 2 =.407 e Refereces [] Joso, N. L., Kotz, S. ad Kemp, A. W. "Uivariate Discrete Distributios". Wiley Series i Probability ad Mathematical Statistics.2 d editio, (992). [2] Miova, L.,"A Geeralizatio of the Classical Discrete Distributios", Compt. Radue Bulg. Acad,Aci, ( 2002)54,2. [3] Kolve, N, Miova, L. ad Neytchev, P., "Iflated Parameter Family of Geeralized Power Series Distributios ad their applicatios i Aalysis of Overdispesed Isurace Data". ARCH, Research Clearig House, 2, (2000). [4]Gapta, L.P. ad Gupta, R. C, Iflatrd Modified Power Series Distributios with applicatios. Commuicatios i Statistics;Theory ad Methods, (995)24(9), [5]Bowers, N. L, Gerber, H. U., Hicma, J. C., Joes, D. A. ad Nesbitt, C. J. Actuarial Mathematics,2 d ed. Schaumburg, III.:Society of Actuaries(997). [6] Rolsi, T. Schmidli, H., Schmidt, V. ad Teugels, J. Stochastic Processes for Isurace ad Fiace, Joh Wiley&Sos, Chichester (999). [7]Wielma, R. Ecoometric Aalysis of Cout Data,3 rd Editio. Spriger-Verlag: Berli (2000). 697