A New Flexible Discrete Distribution: Theory and Empirical Evidence

Transcription

1 A ew Flexible Disrete Distribution: Theor nd Empiril Evidene Abstrt Giovnni Pollio Giovnni De Lu The Prthenope Universit of ples Itl A new disrete distribution with two prmeters is introdued nd disussed. We present its derivtion fter identifing the genertor mehnism of the mss probbilit funtion. Then we show its flexibilit in terms of dispersion index nd show how to estimte the prmeters b mximum likelihood. Finll we ompre it to trditionl s well s flexible disrete distributions using some populr insurne dtsets. The AI nd BI riteri strongl suggest tht the new distributions is ble to provide fit to disrete dt in ver stisftor w.. Introdution The predition of insurne lims is one of the most importnt problems fed in the insurne industr. ompnies need sttistil models ble to provide probbilit lw to the number of lims bsed on disrete distribution given the disrete nture of lims.the hoie of disrete sttistil distribution is ver ritil point beuse the sensitiveness of the predition to the sttistil model n be highl onsiderble.the Poisson nd the egtive Binomil re the most importnt disrete distributions ble to represents the number of lims. However in mn ses these distributions do not provide stisftor results espeill under the irumstne tht high number of zeros ours.the serh for lterntive disrete distributions hs enrihed the literture in remrkble w. The rtile is orgnized s follows. Setion presents the most populr disrete distributions while in Setion 3 we propose novel disrete distribution whih is theoretill ompred to the most trditionl lterntives. Finll in Setion 4 n pplition to some populr insurne dt shows the relevne of the proposl. Some onluding remrks will lose the rtile.. Disrete distributions The nlsis of the most importnt disrete rndom vribles is rried out following unified pproh onsidering the onvergent numeril series with positive terms s the genertor mehnism of the mss probbilit funtion. In prtiulr this pproh n shed light on thereltionship between the probbilit lws for disrete rndom vribles with infinite support nd the numeril series. Atull generl probbilit funtion for disrete rndom vrible Y ould be written s f; θ 0 f; θ where θis the prmeter vetor possibl onstrined to ensure tht f; θ be onvergent with positive vlues nd f denotes mthemtil funtion.. Poissondistribution The Poisson distribution is ertinl the most populr disrete distribution. Its probbilit funtion is given b f; μ f; μ 0 μ! μ 0! e μ μ! In this sethe series with hs positive vlues nd it is strightforwrd to show tht onverges toe μ.it is hrterized b n expeted vlue equl to the vrine so tht the dispersion indexd D σ μ 57

2 ISS Print 9-60 Online enter for Promoting Ides USA tht is the rtion between vrine nd expeted vlue is lws unit. This onstrint is serious drwbk in prtil pplitions.. Poisson Lindledistribution The Poisson-Lindle distribution hs been introdued b Snkrn 970 s disrete distribution for ount dt. It omes from the Poisson distribution with the prmeter μ > 0 following the Lindle distribution with densit funtion f μ; θ + μe θμ θ + with θ > 0. So it belongs to the lss of mixed Poisson distributions. It is interesting to note tht n lterntive genesis of this distribution is given b the following onvergent series with positive vlues funtion of the prmeter 0 : Then f ; P Y. 3 3 f ; 0 0 The dispersion index is given b D θ3 + 4θ + 6θ + θ θ + θ θ + θ + θ3 + 4θ + 6θ + θ θ + θ + As result the index is never below tht is D θ > 0 so the Poisson-Lindle n never dptin se of under-dispersion..3 onw-mxwell Poisson distribution 58 θ The onw-mxwell Poisson MP is generliztion of the Poisson distribution introdued b onw nd Mxwell 96 in queue theor ontext. It depends on two prmeters λ > 0 ndυ > 0.The probbilit funtion is defined s f; θ f; θ 0 λ υ! υ λ 0! υ where θ [λ υ]. In order to ompute this probbilit funtion the normlizing onstnt omputed. A prtil pproh is its truntion fter the k-th term obtining λ 0 f; θ k 0 λ! υ + R k 0 f; θ hs to be where R k k+ is the bsolute error due to the truntion.! υ The dispersion index of the MP distribution n llow for over-dispersion in this se0 < υ < equidispersion υ nd the MP ollpses to Poisson distribution with prmeter nd under-dispersion υ >..4 egtive Binomil Distribution

3 The egtive Binomil distribution depends on two prmeters nd P. The probbilit funtion P P P P P P Y P P P P 0 P P with > 0 nd P > 0. Its bilit to dpt to rel dt finds limittion in the dispersion index D P + P whih is greter thn onl llowing for the se of over-dispersion. 3 A newdistribution A newdisrete distribution is proposed strting from the series with generi term given b + f with R nd 0. It hs positive vlues nd is onvergent.in ft using the rtio riterion we hve f lim lim f is <. Using this onvergent series novel probbilit distribution n be generted tht is f; θ f; θ + whereθ [ ] nd The vlue of depends on the prmetersnd. In prtiulr when inreses the funtion dereses while the reltionship with the prmeters depends on the sign of.when > 0 fixed inreses when inreses. When is negtive the inverse reltionship holds. Figures nd show the shpe of the probbilit funtion: - vring for given vlue of ; - vring for given vlue of. Figure Probbilit funtion with 3 blk 0 red 3 green. 59

4 ISS Print 9-60 Online enter for Promoting Ides USA 60 Figure Probbilit funtion with 3 8 blk red 0.5 green. The omputtion of ompels to trunte the series t high vlue J in the pplition the sum will be omputed up to J The expeted vlue is given b 0 0 Y E It hs no losed form but n be expressed b onvergent series. In ft keeping in mind the rtio riterion the series with generl term given b onverges. Moreover it is es to show tht lim. onsidering tht the vrine is 0 ] [ Y E Y Vr the dispersion index is given b D E Y The index D n ssume vlues both greter thn nd lower thn.the proposed distribution is then ver flexible probbilisti model tht n model both over-dispersion nd under-dispersion. Figure 3 shows the flexible behvior of the dispersion index when prmeter vries given some fixed vlues of.

5 Figure 3 Dispersion index with 3 blk 3 green. In the estimtion step we hve to tke into ount prmeter onstrint whih onerns the prmeter whih hs to be positive.the log-likelihood given b i ln L ln i ln ln i i i ln i ln ln i i i inludes the quntitwhih hs to be onsidered fter trunting the series. The sstem of equtions for the log-likelihood solutions is lnl lnl ln i + i nd is solved using itertive methods. 4 Applitions to rel dtsets ln i + i + i + + i The vrious disrete distributions here onsidered hve been evluted in their bilit to dequtel fit ounting vribles.the nlsis hs been rried out using six utomobiles insurne dtsets relting to different ountries / ers nd onerning the nnul number of lims per poli. The dtsets hs lred been used in Denuit 997. It ontins the dt of four ountries for ers /976 nd 994 Germn for er 960 Switzerlnd for er 96 nd finll Zire for er 974. The min fetures of these dtsets re summrized in Tble. Two remrkble fetures rethe over-dispersion nd the presene of high perentge of zeros from 8% to over 9% for ll the dtsets. The rnge equl to the mximum vlue is low for eh dtset. onsidering the enormous importne in the industril ountries of motor vehiles insurne for ivil libilit towrds third prties in the sttistil nd turil literture speil ttention hs to be pid to the serh for n pproprite probbilisti modelto model the distribution of the number of rod idents in whih motorist hs inurred in given period of time or the nnul number of lims per poli presented to the insurne ompn. The performne of the different distributions hs been evluted using sttistil indies ble to tke into ount the number of prmeters to be estimted the Akike Informtion riterion AI nd the Besin Informtion riterion BI. The AI proposed in Akike 974 is given b AI ln L + r where ln L denotes the log-likelihood nd r is the number of prmeters while the BI Shwrz 978 is BI ln L + rlnn wheren is the number of observtions. The best model is the model presenting the lowest vlue of AI or BI. In Tbles -7θ denotes the unique prmeter in Poisson nd Poisson-Lindel probbilit funtions. For MP egtive Binomil nd novel distribution θ denotes prmeter λ nd respetivel while θ denotes the seond prmeter tht is υ P nd. 6

6 ISS Print 9-60 Online enter for Promoting Ides USA Tble Belgin dt 958 shows tht the Poisson distribution hs ver bd performne nd is ver distnt from the remining distributions in terms of AI nd BI. Aording to the two indies the novel distribution provides the best fit to the dt nd its prinipl ompetitor is the egtive Binomil.Tble 3 Germn dt 960 indites tht ording both to the AI nd to the BI the winner probbilisti model turns out to be the novel distribution. Tble 4 Swiss dt 96 onfirms wht we hve found for the two dtsets tht is the prominent role of the novel distribution whih gin ensures the most stisftor fit of the dt in terms of AI nd BI.In Tble 5 Zirin dt 974 the distribution here proposed provides better results for both the riteri.tbles 6 nd 7 Belgin dt 975/976 nd 994 provideevidenes tht do not devite from the min previous results tht is the best fit is rehed using the novel distribution followed b the egtive Binomil distribution. Finll Tbles 8 nd 9 summrize the AI nd BI vlues Germn 960 Switzerlnd 96 Zire / Observtions Averge Vrine Dispersion index Perentge of 0 s Rnge Tble Desriptive sttistis for the six dtsets θ s. e. θ θ s. e. θ AI BI Poisson Poisson-Lindle MP B ew distribution Tble Estimtion results for Belgindt 958 θ s. e. θ θ s. e. θ AI BI Poisson Poisson-Lindle MP B ew distribution Tble 3 Estimtion results for Germn dt 960 θ s. e. θ θ s. e. θ AI BI Poisson Poisson-Lindle MP B ew distribution Tble 4 Estimtion results for Swiss dt 96 θ s. e. θ θ s. e. θ AI BI Poisson Poisson-Lindle MP B ew distribution Tble 5 Estimtion results for Zirin dt 974

7 θ s. e. θ θ s. e. θ AI BI Poisson Poisson-Lindle MP B ew distribution Tble 6 Estimtion results for Belgin dt 975/976 θ s. e. θ θ s. e. θ AI BI Poisson Poisson-Lindle MP B ew distribution Tble 7 Estimtion results for Belgindt 994 Germn Switzerlnd Zire / Poisson Poisson-Lindle MP B ew distribution Tble 8 Summr of AI riterion 958 Germn 960 Switzerlnd 96 Zire / Poisson Poisson-Lindle MP B ew distribution Tble 9 Summr of BI riterion 5 onlusions In this rtile new disrete distribution is proposed. Its min feture is the flexibilit to dpt to mn dtsets using two prmeters. It llows for dispersion index tking into ount both underdispersion nd overdispersion. The distribution is ompred to trditionl Poisson nd egtive Binomil nd flexible Poisson-Lindle nd MP disrete distribution using populr insurne dtsets lred nlzed in literture. The best fit in terms of AI nd BI is lws hieved with our proposed distribution whih ppers to be ver promising for n disrete dtset. Referenes Akike H. 974 A new look t the sttistil model identifition IEEE Trnstions on Automti ontrol onw R.W. nd Mxwell W.L. 96 A queuing model with stte dependent servie rtes Journl of Industril Engineering 3-36 DenuitM. 997 A new distribution of Poisson-tpe for the number of lims. ASTI Bulletin Snkrn M. 970 The disrete Poisson-Lindle distribution Biometris Shwrz G.E. 978 Estimting the dimension of model Annls of Sttistis