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PredictiveModeling

with

the

TweedieDistribution

GlennMeyers

ISOInnovativeAnalytics

CASAnnual

Meeting

Session

C

25

November16,2009

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Background The

Collective

Risk

Model

Describeasasimulationalgorithm

1.Selectarandomnumberofclaims,N

2.Fori=

1to

N

Selectarandomclaimamount,Zi

3.TotalLoss=1

N

i

i

Z=

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History

1980Discussion

Paper

Program

GlennMeyers

Usedcollective

risk

model

simulation

for

retrospectiverating.

ShawMong

UsedFouriertransformstocalculatecollective

riskmodelprobabilitiesWITHOUTsimulation.

Assumedthat

claim

severity

had

agamma

distribution.

LatergeneralizedbyHeckmanandMeyers(1983)

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History Statistical

Community

(1984)

UniversityofIowaDepartmentofStatistics

Poissonfrequency,

gamma

severity

Usedinorderrestrictedinference

Tweediepublishespaper

Tweedie,M.

C.

K.,

An

Index

which

Distinguishes

betweenSomeImportantExponentialFamilies, inStatistics:ApplicationsandNewDirections,Proceedings

of

the

Indian

Statistical

Golden

Jubilee

InternationalConference,J.K.GhoshandJ.Roy(Eds.),IndianStatisticalInstitute,1984,579604.

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Usesof

Collective

Risk

Model

CalculatingAggregateLossDistributions

Retrospectiverating

Reinsurance

Enterpriseriskmanagement

Fittingmodels

of

insurance

loss

data

Simulationisofnohelpinmaximumlikelihoodestimation

TheTweedie

distribution

is

amember

of

the

exponentialdispersionfamilyandthuscanbeusedinaGLM

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Tweedie

as

a

Compound

Poisson

Model ClaimCountN~Poisson()

ClaimSeverityZ~Gamma(,)KPWLossModelsparameters

TranslateintostandardTweedieparameters

Thisisthesameaspredictedbywellknown

collectiverisk

model

variance

formulas

( )2

12, ,

1 2

pp

pp

+= = =

+

[ ] ( )2 Withsomealgebrawecanseethat

1p

Var Y = = +

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Interpretingthe

Tweedie

p

Forp=

1,

the

Tweedie

is

called

the

OverdispersedPoissondistribution

Claimamountsareconstant

FormostP/Cinsuranceapplications

CV>1whichimpliesp>1.5

2 1, CVforgammadistribution=

1

p +=

+ As , CV 0and 1p

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IllustratingtheEffectofp

Simulationvs

Rs

tweedie package

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IllustratingtheEffectofp

Simulationvs

Rs

tweedie package

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TweedieDensity

Function

Dispersionmodelform DunnandSmyth(2008)

Devianced(y,)

( ) ( ) ( )1

| , , | , , exp ,2

f y p f y p y d y

=

( ) ( ) ( )

2 1 2

, 21 2 1 2

p p p

y yd yp p p p

= +

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GLMswith

the

Tweedie

Distribution

Maximizeloglikelihood MinimizeDeviance

GLMsfocus

only

on

estimating

pandareeithergiven,orestimatedoutsidethe

GLMframework.

Unnecessarytoevaluatef(y|p,y,)VeryfortunateforGLM

NothelpfulformoregeneralmodelsDunnandSmyth(2005,2008)evaluatef(y|p,y,)using

complicatedmath

involving

series

expansions

and

Fourierinversion. Itisalsocomputationallyslow.

DunnistheauthoroftheTweediepackageinR.

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ProblemwiththeCompoundPoisson

Interpretationof

the

Tweedie

Distribution

Aconstantwillforceanartificialrelationshipbetween

the

claim

frequency,

,ortheclaimseverity,. UsesofTweediedistribution

Desireto

build

pure

premium

models

where

claim

frequencyandclaimseverityhavetheirownindependentvariables.

MonteCarloMarkovChainsimulations

Needspeed

)

( )

21 2 1

2 2 2

pp p p

p p p

= = =

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RearrangeCalculationof

TweedieDensity

Objectivesofmethod

Allowtovaryasaninput

Computationallyfast

Keeppfixed

Currentapplications

can

reliably

estimate

p.

Experienceindicatesthatpurepremiumestimatesarerelativelyrobustwithrespecttop.

Becareful

if

accurate

estimates

of

frequency

are

required.

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TweedieDensity

Function

Dispersionmodelform DunnandSmyth(2008)

Firsttermf(y|p,y,)Slowtocalculatewhencalled

Notany

slower

for

calling

with

along

vector

y

( ) ( ) ( )1

| , , | , , exp ,2

f y p f y p y d y

=

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DunnSmythRescaling(2008)

Forgiven,,usetheaboveformulastocalculate,andp.

Findashortandfastapproximationforf(y|p,y,1)

Givenf,

calculate

kand

use

Dunn

Smyth

rescaling

andtheapproximationtocalculatefory>0:

Fory=0,set

( ) ( )2| , , | , , pf y p k f ky p k k =

( ) ( )

= =

1

2

Set

,then

| , , | , ,1

p

k f y p y k f ky p ky

( )

( )

2

0| , , exp

2

p

f p

p

=

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Approximatinglog(f(y|p,y,1))

Calculate, log(f(y|p,y,1))usingdtweedie,ONCE

withalong

vector

yfixed.

Find3valuesinthevectoryfixedthatareclosetoy.

Use

divided

differences

interpolation

to

approximatelog(f(y|p,y,1)).

Increaseaccuracybyputtingmorepointsinyfixed.

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Accuracywith

10,000

yfixeds

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RCode

for

Last

Plot

library(statmod)

library(tweedie)

p=1.5

num=10000log.ybot=log(.01)

log.ytop=log(100)

del=(log.ytoplog.ybot)/num

log.y1=seq(from=log.ybot,to=log.ytop,length=num)

front=dtweedie(exp(log.y1),p,exp(log.y1),1)

#

ldtweedie.front=function(y,lyf,front){

lf=log(front)

ly=log(y)

del=lyf[2]lyf[1]

low=pmax(floor((ly lyf[1])/del),1)

d01=(lf[low+1]lf[low])/del

d12=(lf[low+2]lf[low+1])/del

d23=(lf[low+3]lf[low+2])/del

d012=(d12

d01)/2/deld123=(d23d12)/2/del

d0123=(d123d012)/3/del

ld=lf[low]+(lylyf[low])*d01+(lylyf[low])*(lylyf[low+1])*d012+

(lylyf[low])*(lylyf[low+1])*(lylyf[low+2])*d0123

return(ld)

}

#

ldtweedie.scaled=function(y,p,mu,phi){

dev=y

ll=y

k=(1/phi)^(1/(2p))

ky=k*y

yp=ky>0

dev[yp]=2*((k[yp]*y[yp])^(2p)/((1p)*(2p))k[yp]*y[yp]*

(k[yp]*mu[yp])^(1p)/(1p)+(k[yp]*mu[yp])^(2p)/(2p))

ll[yp]=log(k[yp])+ldtweedie.front(ky[yp],log.y1,front)dev[yp]/2

ll[!yp]=mu[!yp]^(2p)/phi[!yp]/(2p)

return(ll)

}

#

runif(1000,min=0,max=exp(log.ytop))

front.true=log(dtweedie(y,p,y,1))

front.int=ldtweedie.front(y,log.y1,front)

plot(log(y),front.intfront.true)

rug(log(y))

summary(front.intfront.true)

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Remarks

f(y|p,y,)and{i}canbecalculatedinRwiththe

tweediepackage.

logofTweediedensitiescanquicklycalculatedin

closedformusingthecubicapproximation.

Fastcalculationofloglikelihoodisnecessaryfor

BayesianestimationusingtheMCMCsimulations.

Coefficients

of

the

cubic

approximation

allow

for

easycodinginSASandExcelforMLEestimationof.

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Dunn,P.K.&Smyth,G.K.(2008).EvaluationofTweedieexponentialdispersion

modeldensitiesbyFourierinversion.StatisticsandComputing,18,7386.