On an Application of Bayesian Estimation


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1 O a Applicatio of ayesia Estimatio KIYOHARU TANAKA School of Sciece ad Egieerig, Kiki Uiversity, Kowakae, HigashiOsaka, JAPAN EVGENIY GRECHNIKOV Departmet of Mathematics, auma Moscow State Techical Uiversity, Moscow, RUSSIA Abstract:  This paper explais the ayesia versio of estimatio as a method for calculatig credibility premium or credibility umber of claims for shortterm isurace cotracts usig two igrediets: past data o the risk itself ad collateral data from other sources cosidered to be relevat The Poisso/gamma model to estimate the claim frequecy for portfolio of policies ad Normal/ormal model to estimate the pure premium are explaied ad applied Keyword:  Prior distributio, Posterior distributio, ayesia estimator, Poisso/gamma model Itroductio A typical feature of the isurace practice is the eed to set premium at the begiig of the isurace cotract Number of occurrece of claims ad the total claim amouts for isurace compay i the future are the radom evets Their sufficietly precise ad reliable estimate is extremely importat to determie the correct premium for ext year i isurace compay Credibility theory is a techique, or set of techiques, for calculatig premiums for short term isurace cotracts The techique calculates a premium for a risk usig two igrediets: past data from the risk itself ad collateral data, i e data from other sources cosidered to be relevat The essetial features of a credibility premium are that it is a liear fuctio of the past data from the risk itself ad that it allows for the premium to be regularly updated as more data are collected i the future (Waters, 994) A credibility premium represets a compromise betwee the two above metioed sources of iformatio The credibility formula for estimatio of pure premium or claim frequecy P c i ext year is: P Z P + Z () c r ( ) μ where P r is estimatio based o past data from the ow data i isurace compay, or risk, ad μ is estimatio based o collateral data ad Z is a umber betwee zero ad oe, kow as the credibility factor Credibility factor Z is a measure of how much reliace the compay is prepared to place o the data from the policy itself Credibility formula is ofte used i the form P c Z x + ( Z )μ () We will preset ayesia approaches to credibility estimatio by two importat models for isurace practice The ayesia Iferece The ayesia philosophy (763) ivolves a completely differet approach to statistical iferece Suppose x ( x, x, x ) is a radom sample from a populatio specified by desity fuctio f ( x / ) ad it is required to estimate parameter Θ The classical approach to poit estimatio treats parameters as somethig fixed but ukow The essetial differece i the ayesia approach to iferece is that parameters are treated as radom variables ad therefore they have probability distributios Prior iformatio about Θ that we have before collectio of ay data is prior distributio f Θ ( ) that is probability desity fuctio or probability mass fuctio The iformatio about Θ provided by the
2 sample data x x, x, x ) is cotaied i the ( f x i likelihood f ( x / ) ( / ) ayes theorem combies this iformatio with the iformatio cotaied i f Θ ( ) i the form f Θ ( / ) f ( x ) f ( ) ( x ) f ( ) x (3) f d that determies the posterior distributio f Θ ( x) So after collectig appropriate data we determie the posterior distributio that is the basis of all iferece cocerig Θ f x f x f d does ot Note that ( ) ( ) ( ) ivolve Θ It is just a costat eeded to make it a proper desity that itegrates to uity A useful way of expressig the posterior desity is to use proportioality We ca write or simply ( / ) f ( x ) f ( ) f x / (4) posterior likelihood prior The posterior distributio cotais all available iformatio about Θ ad therefore should be used for makig decisios, estimates or ifereces The ayesia approach to estimatio states that we should always start with a prior distributio for ukows parameter, precise or vague accordig to the iformatio available Note that we are referrig to a desity here implyig that Θ is cotiuous This cocers most applicatios because eve whe X is discrete, as i biomial or Poisso distributios, the parameters π or will vary i a cotiuous space 0 ; or 0, + ) respectively There may be some situatios i which we eed oiformative prior For example if Θ is a biomial probability ad we have o prior iformatio about Θ, the uiform distributio o 0 ; as a prior distributio would seem appropriate We ofte have prior iformatio about parameters based o previous practice, respectively, estimates by experts 3 The ayesia estimator If we have foud posterior distributio of a ukow parameter Θ, we eed to aswer the questio how do we use the posterior distributio of Θ, give the sample data x ( x, x, x ), to obtai a estimator of Θ First we must specify the loss fuctio g ( x), which is a measure of the loss icurred whe g ( x) is used as a estimator of Θ We seek a loss fuctio which is zero whe the estimatio is exactly correct, that is g ( x) Θ ad which icreases as g ( x) gets father away from Θ There is oe very commoly used loss fuctio, called quadratic or squared error loss The quadratic loss is defied by L ( g( x) ; ) [ g( x) ] (5) ad it is related to mea square error from classical statistics We will show that the ayesia estimator that arises by miimizig the expected quadratic loss is the mea of posterior distributio So ad ( L( g( x) ; )) [ g( x) ] f ( x) d E E ( L( g( x) ; )) g( x) [ g( x) ] f ( x) d equatig to zero g ( x) f ( x) d f ( x) d ecause of f ( x ) d, we get ( x) E( x) g (6) We will cosider two importat examples of derivatio of the posterior distributio ad the ayesia estimators uder the quadratic error loss for certai estimatio situatios with give prior distributios, importat for isurace practice 4 The Poisso/Gamma Model Suppose we have to estimate the claim frequecy for a risk whose claim umbers have a Poisso distributio with parameter We do ot kow the value of but before havig ay data from risk itself available, we assume that the prior distributio of is a gamma distributio G(α; β) The claim frequecy rate for a class of isurace busiess may lie aywhere betwee 0 ad + A isurer with a large experiece may quite accurately estimate the rate The gamma distributio may be coveiet for represetig ucertaity i a curret estimate of the claim frequecy rate This distributio is over the whole positive rage from 0 to +, ad the mea α/β ca be set equal to the curret best estimate
3 Ucertaity is represeted by variace α/β of the gamma distributio G(α; β) Our objectives is to estimate the ukow parameter Suppose we have past observatios x ( x, x, x ) The ayesia estimate of, with respect to a quadratic loss fuctio, give these data, is E( x) (7) that is the mea of the posterior desity of y assumptio the desity fuctio of the prior G(α; β) distributio is α β α β β α f ( ) e c e (8) Γ( α) The distributio of a umber of claims is the Poisso distributio with a fixed but ukow parameter, so the likelihood fuctio has the form f x (9) xi xi i e c e xi! ( / ) y ayes theorem we get the posterior desity of, give x ( x, x, x ), i the form ( + ) + ( / ) xi β α β α x f x e e e i (0) that is the gamma distributio with ew parameters α α + x i () β β + Thus the ayesia estimator of usig the quadratic loss is α + xi β + which ca be rewritte as α + x x + β + β + α + x β + β β + α β () If we put factor credibility Z (3) β + the E ( x ) Z x + ( Z )μ (4) which is the credibility formula for updatig claim frequecy rates It ca be see from the credibility factor expressio, sice is oegative ad β is positive, that Z is i the rage zero to oe ad it is icreasig fuctio of If o past data from the risk itself are available, the 0 ad Z 0 too ad the best estimate of is α/β, the mea of the prior gamma distributio It ca be see that Z does ot take the value oe for ay fiite value of The value of Z depeds o the amout of data available for the risk, ad the collateral iformatio through β, which reflect the variace α/β of the prior distributio 4 Applicatio of Poisso/Gamma Model The aual umber of claims resultig from motor thirdparty liability isurace i isurace compay i the years is give i Table, colum labelled as x i I the Poisso/gamma model for claim umbers we have assumed our kowledge about the ukow parameter (aual claim rete) is summarized by prior G(α; β) with parameters α 8400 ad β 0,4 Last colum deoted as cotais values of ayes estimators of aual claim rates x i for each year i based of ( i) past observatios by equatio (4) For calculatio of credibility factors Z i we used equatio (3) Table Procedure to update ayes estimate of Year xx i i Z i , , , , , , Source: Ow calculatio 5 The Normal/Normal Model Our problem is to estimate the pure premium, i e the expected aggregate claims for a risk So X is a radom variable deotig total claims from a risk i a comig year ad the distributio of X is ormal, depeds o the value of a ukow parameter Θ The coditioal distributio of X/ is ormal ad the ukow parameter is the mea of this distributio, because of X ~ N( ; ) (5) The prior distributio of is ormal, ~ N ( μ; ) (6) where μ, are kow Suppose we have past observatios of X, x ( x, x, x ) Our problem is to estimate E ( X ) ad we use agai the ayesia estimate with respect to the quadratic loss If was kow, the pure premium would be E ( X ) (7) So the problem of estimatig E ( X ) is the same as the problem of estimatig of as a ayesia estimator
4 ( x) E (8) i e the posterior mea of give x We eed to kow the form of the posterior desity fuctio f ( / x) Suppose we have data of previous observatios x x, x, x ) so we ca express the likelihood ( ( / x) f as ( x / ) ( x ) i x ( x ) i + f e e e As we ca see, the likelihood fuctio is quadratic i, ad ca be show to be proportioal to ( a + a + a3 ) e Whe igorig terms ot ivolvig We ca express the ormal prior distributio as beig proportioal to ( μ ) μ + f ( ) e e π The posterior desity f ( / x) by ayes` theorem is proportioal to ( / x ) f e ad after adjustmets x μ + + e μ + + x + ( / ) e So the posterior distributio ( / x) f x (9) f is a ormal distributio, say with parameters μ ~, ~, i e ( / ) ( ~ μ ) ~ ~ μ + ~ ~ f x c e e (0) We will fid the parameters μ ~, ~ by equatig the power of ad i two differet expressio of f ( / x) The ~ μ + x μ + () ~ + () We ca fid the ayesia estimatio of pure premium as the mea of the posterior distributio, i e μ + x + (3) That ca be rewritte as E( x ) Z x + ( Z )μ (4) which is a credibility estimate of the pure premium E x with factor credibility ( ) Z (5) Applicatio of Normal/Normal Model Total aggregate claims i a particular isurace compay are modelled with a ormal distributio N ( ; ), where is ukow ad Prior iformatio about suggest that it is distributed by N ( μ; ) with kow parameters μ ad Aggregate claims from the last seve years were ot icorporated i the prior iformatio ad they are i Table, colum amed x i The ayes estimatios of the pure premiums for each year by equatio (4) with credibility factors calculated by (5) there are i the last colum of Table Table ayes estimatios of pure premium i x i Z , , , , , , , Source: Ow calculatio 6 Coclusios ayesia estimatio theory provides methods for permaetly updated estimates of the umber of claims ad of the pure premium for each comig year i isurace compay ayesia approach combie prior iformatio that are kow before collected of ay data ad iformatio provided by the sample data, which are umber of claims or aggregate claim amouts i previous years The biggest advatage of the Poisso/gamma model ad Normal/ormal model for isurace practice is possibility to express them i the form of credibility formulas by (4) or (4) These formulas allow easy
5 applicatio i isurace practice, as see from the examples i subsectios 4 ad 5 However, the ayesia approach does have a few serious drawbacks ad limitatios This approach ca be criticized as subjective, because we should always start with a prior distributio of estimated parameters Formulas (3) ad (5) ivolve parameters, β i the former ad, i the latter, which we have assumed to be kow The values of these parameters reflect the subjective opiio of the decisio maker; there is o questio of estimatig these parameters from data The problem of estimatio of ukow parameters whe some data from related risks are available solves the socalled Empirical ayes Credibility Theory, which is ot the subject of this paper Refereces: [] P J olad, Statistical ad Probabilistic Methods i Actuarial Sciece, Lodo: Chapma&Hall/CRC, 007 [] H ühlma, A Gisler, Course i Credibility Theory ad its Applicatios, erli: Spriger, 005 [3] R Kaas, M Goovaerts, J Dhaee, M Deuit, M oder Actuarial Risk Theory, osto: Kluwer Academic Publishers, 00 [4] V Pacáková, The ayesia Iferece i Actuarial Scieces, Cetral Europea Joural for Operatios Research ad Ecoomics, Volume 5, Number 34, 997, pp [5] E Šoltés, V Pacáková, T Šoltésová, Vybraé kredibilé regresé modely v havarijom poisteí, Ekoomický časopis, roč 54, č, 006, pp 688 [6] Y K Tse, Nolife Actuarial Models, Cambridge: Uiversity Press, 009 [7] H R Waters, A Itroductio to Credibility Theory, Lodo ad Ediburgh: Istitute of Actuaries ad Faculty of Actuaries, 994
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