STK 4540Lecture 6
Claim size
The ultimate goal for calculating
the pure premium is pricing
Pure premium = Claim frequency x claim severity
Claim severity 
total claim amount
number of claims
Claim frequency 
number of claims
number of policy years
Parametric and non parametric modelling (section 9.2 EB)
The log-normal and Gamma families (section 9.3 EB)
The Pareto families (section 9.4 EB)
Extreme value methods (section 9.5 EB)
Searching for the model (section 9.6 EB)
2
Non parametric
Claim severity modelling is about
describing the variation in claim size
Log-normal, Gamma
The Pareto
Extreme value
Searching
The graph below shows how claim size varies for fire claims for houses
The graph shows data up to the 88th percentile
How do we model «typical claims» ? (claims that occur regurlarly)
How do we model large claims? (claims that occur rarely)
Claim size fire
700
600
500
Frequency
•
•
•
•
400
300
200
100
0
0
10000
20000
30000
40000
50000
60000
70000
80000
90000
100000 110000 120000 130000 140000 150000
Bin
3
Non parametric
Claim severity modelling is about
describing the variation in claim size
Log-normal, Gamma
The Pareto
Extreme value
Searching
The graph below shows how claim size varies for water claims for houses
The graph shows data up to the 97th percentile
The shape of fire claims and water claims seem to be quite different
What does this suggest about the drivers of fire claims and water claims?
Any implications for pricing?
Claim size water
6000
5000
4000
Frequency
•
•
•
•
•
3000
2000
1000
0
10000 20000 30000 40000 50000 60000 70000 80000 90000 100000 110000 120000 130000 140000 150000
Bin
4
Non parametric
The ultimate goal for calculating the
pure premium is pricing
Log-normal, Gamma
The Pareto
Extreme value
Searching
•
•
•
Claim size modelling can be parametric through families of distributions such as
the Gamma, log-normal or Pareto with parameters tuned to historical data
Claim size modelling can also be non-parametric where each claim zi of the
past is assigned a probability 1/n of re-appearing in the future
A new claim is then envisaged as a random variable Ẑ for which
Pr( Zˆ  zi ) 
•
•
1
, i  1,..., n
n
This is an entirely proper probability distribution
It is known as the empirical distribution and will be useful in Section 9.5.
Size of claim
Client behavour can affect outcome
• Burglar alarm
• Tidy ship (maintenance etc)
• Garage for the car
Here we sample from the
empirical distribution
Where do we draw
the line?
Bad luck
• Electric failure
• Catastrophes
• House fires
Here we use special
Techniques (section 9.5)
5
Non parametric
Example
Log-normal, Gamma
The Pareto
Extreme value
Searching
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
99,1
99,2
99,3
99,4
99,5
99,6
99,7
99,8
99,9
100
45 000
45 301
48 260
50 000
52 580
56 126
60 000
64 219
69 571
74 604
80 000
85 998
95 258
100 000
112 767
134 994
159 646
200 329
286 373
500 000
602 717
662 378
810 787
940 886
1 386 840
2 133 580
2 999 062
3 612 031
4 600 301
8 876 390
120
100
Empirical
distribution
80
60
• The threshold may be set for example at the 99th
percentile, i.e., 500 000 NOK for this product
• The threshold is sometimes called the large
claims threshold
40
20
0
-1 000 000
-
1 000 000
2 000 000
3 000 000
4 000 000
5 000 000
6
Non-parametric modelling
can be useful
•
Scale families of distributions
All sensible parametric models for claim size are of the form
Z  Z 0 , where   0 is a parameter
•
•
and Z0 is a standardized random variable corresponding to   1 .
The large the scale parameter, the more spread out the distribution
Z  Z 0 , Z 0 ~ N (1,1)
  1  Z ~ N (1,1)
  2  Z ~ N (2,2)
  3  Z ~ N (3,3)
7
Non parametric
Scale families of distributions
Log-normal, Gamma
The Pareto
Extreme value
•
All sensible parametric models for claim size are of the form
Searching
Z  Z 0 , where   0 is a parameter
•
•
and Z0 is a standardized random variable corresponding to   1 .
This proportionality is inherited by expectations, standard deviations and
percentiles; i.e. if  0 ,  0 and q0
are expectation, standard devation and

-percentile for Z0, then the same quantities for Z are
  0 ,    0 and q  q 0
• The parameter  can represent for example the exchange rate.
•
•
•
The effect of passing from one currency to another does not change the shape
of the density function (if the condition above is satisfied)
In statistics  is known as a parameter of scale
Assume the log-normal model Z  exp(   ) where  and  are
parameters and  ~ N (0,1) . Then E ( Z )  exp(  (1 / 2) 2 ) . Assume we
rephrase the model as
Z  Z 0 , where Z 0  exp( (1 / 2) 2   ) and   exp(  (1 / 2) 2 )
•
Then
EZ0  E{exp( (1 / 2) 2   )}  E{exp( (1 / 2) 2       )}
 exp( (1 / 2) 2   ) E{exp(    )}  exp( (1 / 2) 2      (1 / 2) 2 )  1 8
Non parametric
Fitting a scale family
•
Log-normal, Gamma
The Pareto
Extreme value
Searching
Models for scale families satisfy
Pr( Z  z )  Pr( Z 0  z /  ) or F(z |  )  F0 (z/  )
•
where F(z |  ) and F0 (z/  ) are the distribution functions of Z and Z0.
Differentiating with respect to z yields the family of density functions
f (z |  ) 
•
dF ( z )
z
f 0 ( ), z  0 where f 0 ( z |  )  0


dz
1
The standard way of fitting such models is through likelihood estimation. If
z1,…,zn are the historical claims, the criterion becomes
n
L(  , f 0 )  n log(  )   log{ f 0 ( zi /  )},
i 1
which is to be maximized with respect to  and other parameters.
9
Shifted distributions
•
•
•
The distribution of a claim may start at some treshold b instead of the origin.
Obvious examples are deductibles and re-insurance contracts.
Models can be constructed by adding b to variables starting at the origin; i.e.
where Z0 is a standardized variable as before. Now
Pr( Z  z )  Pr(b  Z 0  z )  Pr( Z 0 
•
Shifted distributions
z b

)
Example:
• Re-insurance company will pay if claim exceeds 1 000 000 NOK
Z  1000000  Z 0
The payout of the re-insurance company,
given that the total claim amount exceeds 1M
Total claim amount
in NOK
Currency rate for example NOK per EURO,
for example 8 NOK per EURO
10
Non parametric
Shifted distributions
Log-normal, Gamma
The Pareto
Extreme value
Searching
•
•
•
The distribution of a claim may start at some treshold b instead of the origin.
Obvious examples are deductibles and re-insurance contracts.
Models can be constructed by adding b to variables starting at the origin; i.e.
Z  b  Z 0
where Z0 is a standardized variable as before. Now
Pr( Z  z )  Pr(b  Z 0  z )  Pr( Z 0 
and differentiation with respect to z yields
f (z |  ) 
1

f0 (
z b

z b

)
), z  b
which is the density function of random variables with b as a lower limit.
11
Skewness as simple description of shape
•
Skewness
A major issue with claim size modelling is asymmetry and the right tail of the
distribution. A simple summary is the coefficient of skewness
3
  skew( Z )  3 where  3  E ( Z   )3

Negative skewness
Positive skewness
Negative skewness: the left tail is longer; the mass of the distribution
Is concentrated on the right of the figure. It has relatively few low values
Positive skewness: the right tail is longer; the mass of the distribution
Is concentrated on the left of the figure. It has relatively few high values
12
Non parametric
Log-normal, Gamma
Skewness as simple description of shape
The Pareto
Extreme value
Searching
•
A major issue with claim size modelling is asymmetry and the right tail of the
distribution. A simple summary is the coefficient of skewness
3
  skew( Z )  3 where  3  E ( Z   )3

•
The numerator is the third order moment. Skewness should not depend on
currency and doesn’t since
skew( Z ) 
•
•
E ( Z   )3
3
E ( Z 0  0 )3 E ( Z 0   0 )3


 skew( Z 0 )
(  0 )3
 03

Skewness is often used as a simplified measure of shape
The standard estimate of the skewness coefficient
from observations
z1,…,zn is
ˆ 
ˆ 3
s3
n
1
3
where ˆ 
(
z

z
)
 i
n  3  2 / n i 1
3
13
Non parametric
Non-parametric estimation
Log-normal, Gamma
The Pareto
Extreme value
Searching
•
•
The random variable Ẑ that attaches probabilities 1/n to all claims zi of the
past is a possible model for future claims.
Expectation, standard deviation, skewness and percentiles are all closely
related to the ordinary sample versions. For example
n
n
1
E ( Zˆ )   Pr( Zˆ  zi ) zi   zi  z .
i 1
i 1 n
•
Furthermore,
n
n
1
var( Zˆ )  E ( Zˆ  E ( Zˆ ))   Pr( Zˆ  zi )( zi  z )   ( zi  z ) 2
i 1
i 1 n
2
2
n 1
1 n
ˆ
 sd ( Z ) 
s, s 
( zi  z ) 2

n
n - 1 i 1
•
Third order moment and skewness becomes
n
ˆ3 ( Zˆ )
1
3
ˆ
ˆ3 ( Z )   ( zi  z ) and skew( Ẑ) 
n i 1
{sd ( Zˆ )}3
•
•
•
Skewness tends to be small
No simulated claim can be larger than what has been observed in the past
These drawbacks imply underestimation of risk
14
Non parametric
TPL
Log-normal, Gamma
The Pareto
Extreme value
Searching
15
Non parametric
Hull
Log-normal, Gamma
The Pareto
Extreme value
Searching
16
Non parametric
The log-normal family
Log-normal, Gamma
The Pareto
Extreme value
Searching
•
•
A convenient definition of the log-normal
model in the present context is
 2 / 2
as Z  Z 0
where Z 0  e
for  ~ N (0,1)
Mean, standard deviation and skewness are
E ( Z )   , sd(Z)   e
•
2
1
, skew( Z )  (e  2) e
2
2
1
see section 2.4.
Parameter estimation is usually carried out by noting that logarithms are
Gaussian. Thus
Y  log( Z )  log(  )  1 / 2 2  
and when the original log-normal observations z1,…,zn are transformed to
Gaussian ones through y1=log(z1),…,yn=log(zn) with sample mean and
variance y and s y , the estimates of
become
 and 
log( ˆ)  1 / 2ˆ 2  y, ˆ  s y
s /2 y
or ˆ  e y , ˆ  s y .
2
17
Non parametric
Log-normal, Gamma
Log-normal sampling (Algoritm 2.5)
The Pareto
Extreme value
1. Input:  , 
2. Draw U * ~ uniform and
3. Return Z  e  *
Searching
 *   1 (U * )
Lognormal ksi = -0.05 and sigma = 1
70
60
Frequency
50
40
30
20
10
0
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 1,1 1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9 2 2,1 2,2 2,3 2,4 2,5 2,6 2,7 2,8 2,9 3
Bin
18
Non parametric
The lognormal family
Log-normal, Gamma
The Pareto
Extreme value
Searching
•
•
Different choice of ksi and sigma
The shape depends heavily on sigma and is highly skewed when sigma is not
too close to zero
Lognormal ksi = 0.005 and sigma = 0.05
80
70
60
40
30
20
10
Bin
1,15
1,14
1,13
1,12
1,11
1,1
1,09
1,08
1,07
1,06
1,05
1,04
1,03
1,02
1,01
1
0,99
0,98
0,97
0,96
0,95
0,94
0,93
0,92
0,91
0
0,9
Frequency
50
19
Non parametric
The Gamma family
Log-normal, Gamma
The Pareto
Extreme value
Searching
•
The Gamma family is an important family for which the density function is

( /  )  1 x / 
f ( x) 
x e
, x  0, where ( )   x 1e  x dx
( )
0
• It was defined in Section 2.5 as Z  G where G ~ Gamma(  ) is the
standard Gamma with mean one and shape alpha. The density of the standard
Gamma simplifies to

   1 x
f ( x) 
x e , x  0, where ( )   x 1e  x dx
( )
0
Mean, standard deviation and skewness are
E ( Z )   , sd(Z)  /  , skew(Z)  2/ 
and there is a convolution property. Suppose G1,…,Gn are independent with
Gi ~ Gamma( i ) . Then
G ~ Gamma(1  ...   n ) if G 
1G1  ...   nGn
1  ...   n
20
Non parametric
Log-normal, Gamma
Example of Gamma distribution
The Pareto
Extreme value
Searching
1,2
1
0,8
alpha = 1
0,6
alpha = 1,5
alpha = 2,5
0,4
0,2
0,00001
0,15
0,3
0,45
0,6
0,75
0,9
1,05
1,2
1,35
1,5
1,65
1,8
1,95
2,1
2,25
2,4
2,55
2,7
2,85
3
3,15
3,3
3,45
3,6
3,75
3,9
4,05
4,2
4,35
0
21
Non parametric
Log-normal, Gamma
Example: car insurance
The Pareto
Extreme value
Searching
• Hull coverage (i.e., damages on own vehicle in
a collision or other sudden and unforeseen
damage)
• Time period for parameter estimation: 2 years
• Covariates:
–
–
–
–
–
Driving length
Car age
Region of car owner
Tariff class
Bonus of insured vehicle
• 2 models are tested and compared – Gamma
and lognormal
22
Non parametric
Log-normal, Gamma
Comparisons of Gamma and lognormal
The Pareto
Extreme value
Searching
• The models are compared with respect to fit,
results, validation of model, type 3 analysis and QQ
plots
• Fit: ordinary fit measures are compared
• Results: parameter estimates of the models are
compared
• Validation of model: the data material is split in two,
independent groups. The model is calibrated (i.e.,
estimated) on one half and validated on the other
half
• Type 3 analysis of effects: Does the fit of the model
improve significantly by including the specific
variable?
23
Non parametric
Comparison of Gamma and
lognormal - fit
Log-normal, Gamma
The Pareto
Extreme value
Searching
Gamma fit
Criterion
Deviance
Scaled
Deviance
Pearson ChiSquare
Scaled
Pearson X2
Log Likelihood
Full Log
Likelihood
AIC (smaller is
better)
AICC (smaller
is better)
BIC (smaller is
better)
Lognormal fit
Deg. fr.
Verdi
Value/DF
546
12 926,1628
23,6743
546
669,2070
1,2257
546
7 390,8283
13,5363
546
382,6344
0,7008
_
- 5 278,7043
_
_
- 5 278,7043
_
_
10 595,4086
_
_
10 596,8057
_
_
10 677,7747
_
Criterion
Deviance
Scaled
Deviance
Pearson ChiSquare
Scaled
Pearson X2
Log Likelihood
Full Log
Likelihood
AIC (smaller is
better)
AICC (smaller
is better)
BIC (smaller is
better)
Deg. fr.
Verdi
Value/DF
2 814
119 523,2128
42,4745
2 814
2 838,0000
1,0085
2 814
119 523,2128
42,4745
2 814
2 838,0000
1,0085
_
- 7 145,8679
_
_
- 7 145,8679
_
_
14 341,7357
_
_
14 342,1980
_
_
14 490,5071
_
24
Non parametric
Comparison of Gamma
and lognormal – type 3
Log-normal, Gamma
The Pareto
Extreme value
Searching
Gamma fit
Source
Lognormal fit
Deg. fr.
Chi-square
Tariff class
5
51,75
<.0001 LR
<.0001 LR
Bonus
2
177,74
<.0001 LR
7
48,14
<.0001 LR
Deg. fr.
Chi-square
Pr>Chi-sq Method
Tariff class
5
70,75
<.0001 LR
Bonus
2
19,32
Source
Pr>Chi-sq Method
Region
7
20,15
0,0053 LR
Region
Car age
3
342,49
<.0001 LR
Driving length
6
70,18
<.0001 LR
Car age
3
939,46
<.0001 LR
25
Non parametric
QQ plot Gamma model
Log-normal, Gamma
The Pareto
Extreme value
Searching
26
Non parametric
QQ plot log normal model
Log-normal, Gamma
The Pareto
Extreme value
Searching
27
Non parametric
Log-normal, Gamma
The Pareto
Extreme value
Searching
70 000
250,0 %
Results tariff class
60 000
200,0 %
50 000
150,0 %
40 000
Risk years
Difference from reference,
gamma model
30 000
100,0 %
Difference from reference,
lognormal model
20 000
50,0 %
10 000
0
0,0 %
1
2
3
4
5
6
28
Non parametric
Log-normal, Gamma
The Pareto
Extreme value
Searching
160 000
120,0 %
Results bonus
140 000
100,0 %
120 000
80,0 %
100 000
Risk years
80 000
60,0 %
60 000
40,0 %
Difference from reference,
gamma model
Difference from reference,
lognormal model
40 000
20,0 %
20 000
0
0,0 %
70,00 %
75,00 %
Under 70%
29
Non parametric
Log-normal, Gamma
The Pareto
Extreme value
Searching
60 000
50 000
40 000
Results region
140,0 %
120,0 %
100,0 %
80,0 %
30 000
20 000
10 000
0
60,0 %
Risk years
40,0 %
Difference from reference,
gamma model
20,0 %
0,0 %
Difference from reference,
lognormal model
30
Non parametric
Log-normal, Gamma
The Pareto
Extreme value
Searching
120 000
120,0 %
Results car age
100 000
100,0 %
80 000
80,0 %
Risk years
60 000
60,0 %
40 000
40,0 %
20 000
20,0 %
0
Difference from reference,
gamma model
Difference from reference,
lognormal model
0,0 %
<= 5 years 5-10 years
10-15
years
>15 years
31
Non parametric
Log-normal, Gamma
The Pareto
Extreme value
Searching
80,00
80,00
Validation bonus
70,00
70,00
60,00
60,00
50,00
50,00
40,00
40,00
Validation region
30,00
30,00
20,00
20,00
10,00
10,00
Difference Gamma
Difference lognormal
0,00
0,00
Total
70 %
75 %
Below 70%
32
Non parametric
Log-normal, Gamma
The Pareto
Extreme value
Searching
100,00
Validation car age
90,00
70,00
Validation tariff class
60,00
80,00
70,00
50,00
60,00
40,00
50,00
Difference Gamma
40,00
30,00
30,00
20,00
Difference lognormal
20,00
10,00
10,00
0,00
Total
<= 5 years
5-10years
10-15years
>15 years
0,00
Total
1
2
3
4
5
6
33
Non parametric
Conclusions so far
Log-normal, Gamma
The Pareto
Extreme value
Searching
• None of the models are very good
• Lognormal behaves worst and can be
discarded
• Can we do better?
• We try Gamma once more, now exluding
the 0 claims (about 17% of the claims)
– Claims where the policy holder has no guilt
(other party is to blame)
34
Non parametric
Comparison of Gamma
and lognormal - fit
Log-normal, Gamma
The Pareto
Extreme value
Searching
Gamma fit
Criterion
Deviance
Scaled
Deviance
Pearson ChiSquare
Scaled
Pearson X2
Log Likelihood
Full Log
Likelihood
AIC (smaller is
better)
AICC (smaller
is better)
BIC (smaller is
better)
Gamma without zero claims fit
Deg. fr.
Verdi
Value/DF
546
12 926,1628
23,6743
546
669,2070
1,2257
546
7 390,8283
13,5363
546
382,6344
0,7008
_
- 5 278,7043
_
_
- 5 278,7043
_
_
10 595,4086
_
_
10 596,8057
_
_
10 677,7747
_
Criterion
Deviance
Scaled
Deviance
Pearson ChiSquare
Scaled
Pearson X2
Log Likelihood
Full Log
Likelihood
AIC (smaller is
better)
AICC (smaller
is better)
BIC (smaller is
better)
Deg. fr.
Verdi
Value/DF
494
968,9122
1,9614
494
546,4377
1,1061
494
949,1305
1,9213
494
535,2814
1,0836
_
- 5 399,8298
_
_
- 5 399,8298
_
_
10 837,6596
_
_
10 839,2043
_
_
10 918,1877
_
35
Comparison of Gamma
and lognormal – type 3
Gamma fit
Source
Non parametric
Log-normal, Gamma
The Pareto
Extreme value
Searching
Gamma without zero claims fit
Deg. fr.
Chi-square
Pr>Chi-sq Method
Tariff class
5
70,75
<.0001 LR
Bonus
2
19,32
<.0001 LR
Region
7
20,15
0,0053 LR
Car age
3
342,49
<.0001 LR
Source
Deg. fr.
Chi-square
Pr>Chi-sq Method
BandCode1
5
101,22
<.0001 LR
CurrNCD_Cd
KundeFylkeNav
n
2
43,04
<.0001 LR
7
48,08
<.0001 LR
Side1Verdi6
3
70,76
<.0001 LR
36
Non parametric
QQ plot Gamma
Log-normal, Gamma
The Pareto
Extreme value
Searching
37
QQ plot Gamma model
without zero claims
Non parametric
Log-normal, Gamma
The Pareto
Extreme value
Searching
38
Non parametric
Log-normal, Gamma
The Pareto
Extreme value
Searching
70 000
250,0 %
Results tariff class
60 000
200,0 %
50 000
Risk years
150,0 %
40 000
Difference from reference,
gamma model
30 000
100,0 %
Difference from reference,
Gamma model without zero
claims
20 000
50,0 %
10 000
0
0,0 %
1
2
3
4
5
6
39
Non parametric
Log-normal, Gamma
The Pareto
Extreme value
Searching
160 000
140,0 %
Results bonus
140 000
120,0 %
120 000
100,0 %
Risk years
100 000
80,0 %
Difference from reference,
gamma model
80 000
60,0 %
60 000
40,0 %
40 000
Difference from reference,
Gamma model without zero
claims
20,0 %
20 000
0
0,0 %
70,00 %
75,00 %
Under 70%
40
Non parametric
Log-normal, Gamma
The Pareto
Extreme value
Searching
60 000
50 000
40 000
Results region
140,0 %
120,0 %
100,0 %
80,0 %
30 000
Risk years
60,0 %
20 000
10 000
0
40,0 %
Difference from reference,
gamma model
20,0 %
0,0 %
Difference from reference,
Gamma model without zero
claims
41
Non parametric
Log-normal, Gamma
The Pareto
Extreme value
Searching
120 000
120,0 %
Results car age
100 000
100,0 %
80 000
80,0 %
Risk years
60 000
60,0 %
Difference from reference,
gamma model
40 000
40,0 %
Difference from reference,
Gamma model without zero
claims
20 000
20,0 %
0
0,0 %
<= 5 years 5-10 years
10-15
years
>15 years
42
Non parametric
Log-normal, Gamma
The Pareto
Extreme value
Searching
80,00
70,00
Validation region
60,00
50,00
40,00
30,00
Difference Gamma
20,00
Difference lognormal
10,00
0,00
12,00
12,00
Validation bonus
Validation region
10,00
10,00
8,00
8,00
6,00
6,00
4,00
4,00
2,00
0,00
2,00
Difference Gamma
Difference Gamma without
zeroes
0,00
Total
70 %
75 %
Below 70%
43
Non parametric
Log-normal, Gamma
The Pareto
Extreme value
Searching
70,00
Validation tariff class
60,00
50,00
40,00
Difference Gamma
30,00
Difference lognormal
20,00
10,00
0,00
12,00
Total
Validation car age
1
2
40,00
10,00
4
5
6
Validation tariff class
35,00
8,00
3
30,00
25,00
6,00
Difference Gamma
20,00
4,00
Difference Gamma
without zeroes
15,00
10,00
2,00
5,00
0,00
0,00
Total
<= 5 years
5-10years
10-15years
>15 years
Total
1
2
3
4
5
6
44
Backup slides
Non parametric
Fitting a scale family
•
Log-normal, Gamma
The Pareto
Extreme value
Searching
Models for scale families satisfy
Pr( Z  z )  Pr( Z 0  z /  ) or F(z |  )  F0 (z/  )
•
where F(z |  ) and F0 (z/  ) are the distribution functions of Z and Z0.
Differentiating with respect to z yields the family of density functions
f (z |  ) 
•
dF ( z )
z
f 0 ( ), z  0 where f 0 ( z |  )  0


dz
1
The standard way of fitting such models is through likelihood estimation. If
z1,…,zn are the historical claims, the criterion becomes
n
L(  , f 0 )  n log(  )   log{ f 0 ( zi /  )},
i 1
•
•
•
which is to be maximized with respect to  and other parameters.
A useful extension covers situations with censoring.
Perhaps the situation where the actual loss is only given as some lower bound
b is most frequent.
Example:
• travel insurance. Expenses by loss of tickets (travel documents) and
passport are covered up to 10 000 NOK if the loss is not covered by any
of the other clauses.
46
Non parametric
Fitting a scale family
Log-normal, Gamma
The Pareto
Extreme value
Searching
•
The chance of a claim Z exceeding b is 1  F0 (b /  ) , and for nb such events
with lower bounds b1,…,bnb the analogous joint probability becomes
{1  F0 (b1 /  )}x...x{1  F0 (bnb /  )}.
Take the logarithm of this product and add it to the log likelihood of the fully
observed claims z1,…,zn. The criterion then becomes
n
nb
i 1
i 1
L(  , f 0 )  n log(  )   log{ f 0 ( zi /  )}   log{1  F0 ( zi /  )},
complete information
censoring to the right
47