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![Page 1: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/1.jpg)
On Predictive Modeling for Claim Severity
Glenn MeyersISO Innovative Analytics
CARe SeminarJune 6-7, 2005
![Page 2: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/2.jpg)
Problems with Experience Rating for
Excess of Loss Reinsurance
• Use submission claim severity data– Relevant, but– Not credible– Not developed
• Use industry distributions– Credible, but– Not relevant (???)
![Page 3: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/3.jpg)
General Problems withFitting Claim Severity Distributions
• Parameter uncertainty– Fitted parameters of chosen model are
estimates subject to sampling error.
• Model uncertainty– We might choose the wrong model. There is
no particular reason that the models we choose are appropriate.
• Loss development– Complete claim settlement data is not always
available.
![Page 4: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/4.jpg)
Outline of Remainder of Talk
• Quantifying Parameter Uncertainty– Likelihood ratio test
• Incorporating Model Uncertainty– Use Bayesian estimation with likelihood
functions– Uncertainty in excess layer loss estimates
• Bayesian estimation with prior models based on data reported to a statistical agent– Reflect insurer heterogeneity– Develops losses
![Page 5: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/5.jpg)
How Paper is Organized
• Start with classical hypothesis testing.– Likelihood ratio test
• Calculate a confidence region for parameters.• Calculate a confidence interval for a function
of the parameters.– For example, the expected loss in a layer
• Introduce a prior distribution of parameters.• Calculate predictive mean for a function of
parameters.
![Page 6: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/6.jpg)
The Likelihood Ratio Test
1Let ( ,..., ) be a set of observed
losses.nx xx
1Let ( ,..., ) be a parameter vector
for your chosen loss model.kp pp
ˆLet be the maximum likelihood
estimate of given .
p
p x
![Page 7: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/7.jpg)
The Likelihood Ratio Test
0 1Test H : against H : * *p p p p
0
*
2
Theorem 2.10 in Klugman, Panjer & Willmot
If H is true then:
ˆ ln 2 ln ; ln ;
has a distribution with degrees
of freedom.
LR L p x L p x
k
2Use distribution to find critical values.
![Page 8: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/8.jpg)
An Example – The Pareto Distribution
( ) 1F xx
• Simulate random sample of size 1000
= 2.000, = 10,000
Maximum Likelihood = -10034.660 with
ˆˆ 8723.04 1.80792
![Page 9: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/9.jpg)
Hypothesis Testing Example
• Significance level = 5%
2 critical value = 5.991
• H0: () = (10000, 2)
• H1: () ≠ (10000, 2)
• lnLR = 2(-10034.660 + 10035.623) =1.207
• Accept H0
![Page 10: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/10.jpg)
Hypothesis Testing Example
• Significance level = 5%
2 critical value = 5.991
• H0: () = (10000, 1.7)
• H1: () ≠ (10000, 1.7)
• lnLR = 2(-10034.660 + 10045.975) =22.631
• Reject H0
![Page 11: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/11.jpg)
Confidence Region
• X% confidence region corresponds to the 1-X% level hypothesis test.
• The set of all parameters () that fail to reject corresponding H0.
• For the 95% confidence region:– (10000, 2.0) is in.– (10000, 1.7) out.
![Page 12: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/12.jpg)
Confidence Region
Outer Ring 95%, Inner Ring 50%
0.0
0.5
1.0
1.5
2.0
2.5
0 5000 10000 15000Theta
Alp
ha
![Page 13: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/13.jpg)
Grouped Data
• Data grouped into four intervals– 562 under 5000– 181 between 5000 and 10000– 134 between 10000 and 20000– 123 over 20000
• Same data as before, only less information is given.
![Page 14: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/14.jpg)
Confidence Region for Grouped Data
Outer Ring 95%, Inner Ring 50%
0.0
0.5
1.0
1.5
2.0
2.5
0 5000 10000 15000Theta
Alp
ha
![Page 15: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/15.jpg)
Confidence Region for Ungrouped Data
Outer Ring 95%, Inner Ring 50%
0.0
0.5
1.0
1.5
2.0
2.5
0 5000 10000 15000Theta
Alp
ha
![Page 16: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/16.jpg)
Estimation with Model UncertaintyCOTOR Challenge – November 2004
• COTOR published 250 claims– Distributional form not revealed to participants
• Participants were challenged to estimate the cost of a $5M x $5M layer.
• Estimate confidence interval for pure premium
![Page 17: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/17.jpg)
You want to fit a distribution to 250 Claims
• Knee jerk first reaction, plot a histogram.
0 1 2 3 4 5 6 7
x 106
0
50
100
150
200
250
Claim Amount
Cou
nt
Histogram of Cotor Data
![Page 18: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/18.jpg)
This will not do! Take logs• And fit some standard distributions.
6 7 8 9 10 11 12 13 14 15 160
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Log of Claim Amounts
Den
sity
lcotor data
lognormal
gamma
Weibull
![Page 19: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/19.jpg)
Still looks skewed. Take double logs.
• And fit some standard distributions.
1.8 2 2.2 2.4 2.6 2.80
0.5
1
1.5
2
2.5
log log of Claim Amounts
Den
sity
llcotor data
Lognormal
Gamma
Weibull
![Page 20: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/20.jpg)
Still looks skewed. Take triple logs.• Still some skewness. • Lognormal and gamma fits look somewhat better.
0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10
1
2
3
4
5
Triple log of Claim Amounts
Den
sity
lllcotor data
Lognormal
Gamma
Normal
![Page 21: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/21.jpg)
Candidate #1Quadruple lognormal
Distribution: Lognormal Log likelihood: 283.496 Domain: 0 < y < Inf Mean: 0.738351 Variance: 0.006189 Parameter Estimate Std. Err. Mu -0.30898 0.00672 sigma 0.106252 0.004766 Estimated covariance of parameter estimates: mu sigma Mu 4.52E-05 1.31E-19 Sigma 1.31E-19 2.27E-05
![Page 22: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/22.jpg)
Candidate #2Triple loggamma
Distribution: Gamma Log likelihood: 282.621 Domain: 0 < y < Inf Mean: 0.738355 Variance: 0.00615 Parameter Estimate Std. Err. A 88.6454 7.91382 B 0.008329 0.000746 Estimated covariance of parameter estimates: a b A 62.6286 -0.00588 B -0.00588 5.56E-07
![Page 23: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/23.jpg)
Candidate #3Triple lognormal
![Page 24: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/24.jpg)
All three cdf’s are within confidence interval for the quadruple lognormal.
0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Triple log of Claim Amounts
Cum
ulat
ive
prob
abili
ty
lllcotor data
Lognormal
confidence bounds (Lognormal)
Gamma
Normal
![Page 25: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/25.jpg)
Elements of Solution
• Three candidate models– Quadruple lognormal– Triple loggamma– Triple lognormal
• Parameter uncertainty within each model• Construct a series of models consisting of
– One of the three models.– Parameters within a broad confidence interval
for each model. – 7803 possible models
![Page 26: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/26.jpg)
Steps in Solution
• Calculate likelihood (given the data) for each model.
• Use Bayes’ Theorem to calculate posterior probability for each model– Each model has equal prior probability.
Posterior model|data Likelihood data|model Prior model
![Page 27: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/27.jpg)
Steps in Solution
• Calculate layer pure premium for 5 x 5 layer for each model.
• Expected pure premium is the posterior probability weighted average of the model layer pure premiums.
• Second moment of pure premium is the posterior probability weighted average of the model layer pure premiums squared.
![Page 28: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/28.jpg)
CDF of Layer Pure Premium
Probability that layer pure premium ≤ x
equals
Sum of posterior probabilities for which the
model layer pure premium is ≤ x
![Page 29: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/29.jpg)
Numerical Results
Mean 6,430 Standard Deviation 3,370 Median 5,780
Range Low at 2.5% 1,760 High at 97.5% 14,710
![Page 30: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/30.jpg)
Histogram of Predictive Pure Premium
Predictive Distribution of the Layer Pure Premium
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Low End of Amount (000)
Den
sity
![Page 31: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/31.jpg)
Example with Insurance Data
• Continue with Bayesian Estimation
• Liability insurance claim severity data
• Prior distributions derived from models based on individual insurer data
• Prior models reflect the maturity of claim data used in the estimation
![Page 32: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/32.jpg)
Initial Insurer Models
• Selected 20 insurers– Claim count in the thousands
• Fit mixed exponential distribution to the data of each insurer
• Initial fits had volatile tails
• Truncation issues– Do small claims predict likelihood of large
claims?
![Page 33: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/33.jpg)
Initial Insurer Models
0
5,000
10,000
15,000
20,000
25,000
30,000
35,000
40,000
45,000
1,000 10,000 100,000 1,000,000 10,000,000
Loss Amount - x
Lim
ited
Ave
rage
Sev
erit
y
![Page 34: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/34.jpg)
Low Truncation Point
0
500
1,000
1,500
2,000
2,500
3,000
3,500
4,000
4,500
5,000
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
Probability That Loss is Over 5,000
500
x 5
00 L
ayer
Ave
rage
Sev
erit
y
![Page 35: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/35.jpg)
High Truncation Point
0
500
1,000
1,500
2,000
2,500
3,000
3,500
4,000
4,500
5,000
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07
Probability That Loss is Over 100,000
500
x 5
00 L
ayer
Ave
rage
Sev
erit
y
![Page 36: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/36.jpg)
Selections Made
• Truncation point = $100,000
• Family of cdf’s that has “correct” behavior– Admittedly the definition of “correct” is
debatable, but– The choices are transparent!
![Page 37: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/37.jpg)
Selected Insurer Models
0
5,000
10,000
15,000
20,000
25,000
30,000
35,000
40,000
45,000
100,000 1,000,000 10,000,000
Loss Amount - x
Lim
ited
Ave
rage
Sev
erit
y
![Page 38: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/38.jpg)
Selected Insurer Models
0
1,000
2,000
3,000
4,000
5,000
6,000
0.00 0.01 0.01 0.02 0.02 0.03 0.03 0.04 0.04 0.05
Probability That Loss is Over 100,000
500
x 50
0 L
ayer
Ave
rage
Sev
erit
y
![Page 39: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/39.jpg)
Each model consists of
1. The claim severity distribution for all claims settled within 1 year
2. The claim severity distribution for all claims settled within 2 years
3. The claim severity distribution for all claims settled within 3 years
4. The ultimate claim severity distribution for all claims
5. The ultimate limited average severity curve
![Page 40: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/40.jpg)
Three Sample Insurers Small, Medium and Large
• Each has three years of data
• Calculate likelihood functions– Most recent year with #1 on prior slide– 2nd most recent year with #2 on prior slide– 3rd most recent year with #3 on prior slide
• Use Bayes theorem to calculate posterior probability of each model
![Page 41: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/41.jpg)
Formulas for Posterior Probabilities
, 1 ,
, ,, 11
AY m i AY m ii AY m
AY m
F x F xP
F x
,9 3
, ,1 1
i AYn
m i AY mi AY
l P
Posterior( ) Prior( )mm l m
Model (m) Cell Probabilities
Likelihood (m)
Using Bayes’ Theorem
Number of claims
![Page 42: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/42.jpg)
ResultsTaken from
paper.
IntervalLower Claim Prior Posterior $500K x $1M x
Lags Bound Count Model # Probability $500K $1M1 100,000 15 1 0.016406 763 5411 200,000 2 2 0.041658 911 6451 300,000 1 3 0.089063 1,153 6821 400,000 2 4 0.130281 1,224 7961 500,000 0 5 0.157593 1,281 9121 750,000 0 6 0.110614 1,390 9781 1,000,000 0 7 0.075702 1,494 1,0401 1,500,000 0 8 0.053226 1,587 1,0951 2,000,000 0 9 0.080525 1,849 1,328
10 0.104056 2,069 1,52311 0.129925 2,417 1,828
1-2 100,000 40 12 0.010896 2,598 1,9161-2 200,000 10 13 0.000007 2,788 1,9221-2 300,000 1 14 0.000009 3,004 2,1241-2 400,000 0 15 0.000011 3,202 2,3091-2 500,000 2 16 0.000013 3,382 2,4771-2 750,000 0 17 0.000014 3,543 2,6281-2 1,000,000 2 18 0 4,058 3,2111-2 1,500,000 0 19 0 4,663 3,7841-2 2,000,000 0 20 0 5,354 4,440
1,572 1,1131-3 100,000 76 463 3851-3 200,000 261-3 300,000 111-3 400,000 31-3 500,000 81-3 750,000 01-3 1,000,000 01-3 1,500,000 01-3 2,000,000 0
Posterior MeanPosterior Std. Dev.
Exhibit 1 – Small InsurerLayer Pure Premium
![Page 43: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/43.jpg)
Formulas for Ultimate Layer Pure Premium
• Use #5 on model (3rd previous) slide to calculate ultimate layer pure premium
20
=1
202 2
=1
Posterior Mean = Layer Pure Premium( ) Posterior( ).
Posterior Standard Deviation =
Layer Pure Premium( ) Posterior( ) Posterior Mean .
m
m
m m
m m
![Page 44: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/44.jpg)
Results
• All insurers were simulated from same population.
• Posterior standard deviation decreases with insurer size.
$500K x $1M x $500K x $1M x $500K x $1M x$500K $1M $500K $1M $500K $1M
1,572 1,113 1,344 909 1,360 966463 385 278 245 234 188
Small Insurer Medium Insurer Large Insurer
Posterior MeanPosterior Std. Dev.
Layer Pure PremiumLayer Pure Premium Layer Pure Premium
![Page 45: On Predictive Modeling for Claim Severity Glenn Meyers ISO Innovative Analytics CARe Seminar June 6-7, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022070400/56649f145503460f94c28d8a/html5/thumbnails/45.jpg)
Possible Extensions
• Obtain model for individual insurers
• Obtain data for insurer of interest
• Calculate likelihood, Pr{data|model}, for each insurer’s model.
• Use Bayes’ Theorem to calculate posterior probability of each model
• Calculate the statistic of choice using models and posterior probabilities– e.g. Loss reserves