An original paper which suggests a way of thinking for semilinear credibility theory develpment, founded on analysis of the functions of the observable random variables. This line of thought fits perfectly within the framework of the greatest accuracy credibility theory.linear functions, the transformed observations, semi-linear credibility estimators

Virginia ATANASIU

English

Research Papers in Economics

Informatica Economică, nr. 1 (41)/2007  117
Semi-linear Credibility Results 
 
Virginia ATANASIU 
Academy of Economic Studies, Bucharest 
 
An original paper which suggests a way of thinking for semi-linear credibility theory devel-
opment, founded on analysis of the functions of the observable random variables. 
This line of thought fits perfectly within the framework of the greatest accuracy credibility 
theory. 
Key words: linear functions, the transformed observations, semi-linear credibility estimators. 
 
ntroduction 
Se
ear f
mi-linear credibility estimators are lin-
unctions of transformed observations. 
The estimators mainly considered here - in 
the first section - are linear functions of sev-
eral functions f1,...,fr of the observable ran-
dom variables. So instead of considering lin-
ear combinations of the observable variables 
themselves, one could consider as estimators 
the class of linear combinations of given 
functions of the observable variables, and 
solve - see the second section-: 
MinE   () ( ) ⎪
⎬
⎫ ⎪
⎨
⎧ ⎤ ⎡
−∑∑
2
n t
X f c X f
⎪ ⎭ ⎪ ⎩
⎥
⎦
⎢
⎣ ==
+
11
1 0
p r
r pr pr t
In this way one obtains semi-linear credibil-
ity results. One may either assume the func-
tions f0, fpr to be given in advance, or one 
may try to determine the best choice. If one 
wants to estimate a variance-like term, it 
might be appropriate to consider a quadratic 
function f1 = fpr. So, in case one would like to 
estimate the variance, a quadratic function is 
considered. Probably it is also better to take 
quadratic functions of the observable vari-
ables than to approximate by a combination 
of linear functions. This is also more in har-
mony with the dimensions of the problem. In 
some cases one only has data on large claims, 
so one takes x, if x >α , 0 otherwise as the 
claim amount. One possible choice for f is 
min(x,d), where d denotes the threshold 
value above which a claim is called "large". 
This special choice enables us to evaluate the 
effect of reinsurance on the risk premium. 
Sometimes the results of credibility are too 
sensitive to changes in large claims. Choos-
ing f like this gives us the possibility to avoid 
too large fluctuations in the premiums. This 
choice should be considered in combination 
with an excess of loss reinsurance treaty with 
retention d for future operations. This article 
is devoted to semi-linear credibility, where 
one examines functions of the random vari-
ables representing claim amounts, rather than 
the claim amounts themselves. 
 
Several approximating functions 
Here and in the following we present the 
main results leaving the detailed computa-
tions to the reader. Consider a finite se-
quenceθ , X1,..., Xt, Xt+1 of random variables. 
Assume that for fixedθ , the variables X1,..., 
Xt+1 are conditionally independent and iden-
tically distributed (i.i.d.). The variables 
X1,...,Xt are observable, and θ  is the struc-
ture variable. The variable Xt+1 is considered 
as being not (yet) observable. We assume 
that fp(Xr), p= 1 , 1 , , + = t r n 0  have finite vari-
ance. For f0 we take the function of Xt+1 we 
want to forecast. We use the notation: 
( ) = E[fp(Xr)| θ μ p θ ]   (1.1) 
(p = )  1 , 1 , , 0 + = t r n
This expression does not depend on r. Our 
problem is the determination of the linear 
combination of 1 and the random variables: 
fp(Xr) (p =  t r n , 1 , , = 1 ) (1.2) 
closest to ( )( ) [ ] θ θ μ | 1 0 0 + = t X f E  in the least 
squares sense. It is equivalent to determine 
the linear combination of that form closest to 
f0(Xt+1). For this model we define the follow-
ing structure parameters: 
mp=E ( ) [ ] [ ] { } ( ) [ ] r p r p p X f E X f E E = = θ θ μ | ) (  
(1.3) 
apq =E
I 
[ ( )()] { } θ | , r q r p X f X f Cov  (1.4) Informatica Economică, nr. 1 (41)/2007  118 
bpq = Cov () () [ ] θ μ θ μ q p ,  (1.5) 
cpq =Cov ()() [ ] r q r p X f X f ,  (1.6) 
dpq =Cov ()( ) [ ] θ μq r p X f ,  (1.7) 
for p, q = n , 0 . These expressions do not de-
pend on r = 1 , 1 + t . The structure parameters 
are connected by the following relations: 
cpq = apq + bpq   (1.8) 
dpq = bpq (1.9) 
for p, q = n , 0 . This follows from the covari-
ance relations obtained in the probability 
theory, where they are very well-known. In-
deed, we have: 
cpq = Cov  ()() [ ] ( ) [ { ( ) ]} θ | , , r q r p r q r p X f X f Cov E X f X f = +Cov{E [fp(Xr)|θ ], 
, E[fq(Xr)| ]} θ  = apq+Cov ( ) ( ) [ ] ( ) n q p b a pq pq q p , 0 , , = + = θ μ θ μ  
So the verification of equality (1.8) is readily performed. Next, we have: 
dpq = Cov ()( ) [ ] ( ) ( ) [ ] { } θ θ μ θ μ | , , q r p q r p X f Cov E X f = +Cov{E[fp(Xr)|θ ], 
[ ]} θ θ μ | ) ( q E  = E () [ ] { } [ ] ( ) [ ]} θ θ μ θ θ θ μ | | ) ( | ) ( q r p q r p E X f E X f E − + 
+Cov () ( ) [ ] { () ( ) [ ] ( ) ( )} pq q p r p q q p b X f E E + − = θ μ θ μ θ θ μ θ μ θ μ | , = E[ () θ μq  
() ( )( ) ] ( ) ) , 0 , ( , 0 0 n q p b b b E b pq pq pq pq q p p = = + = + = + − ⋅ θ μ θ μ θ μ
Therefore the verification of equality (1.9) is 
readily performed. 
 
Optimal non-homogeneous linearized es-
timator 
Just as in the case of considering linear com-
binations of the observable variables them-
selves, we can also obtain non-homogeneous 
credibility estimates, taking as estimators the 
class of linear combinations of given func-
tions of the observable variables, as shown in 
the following theorem: 
Theorem 2.1 (optimal non-homogeneous 
linearized estimator) 
The linear combination of 1 and the random 
variables fp(Xr) (p =  t r n , 1 , , 1 = ) closest to 
() ( ) [] θ θ μ | 1 0 0 + = t X f E  and to f0(Xt+1) in the 
least squares sense equals: 
M =  p
n
p
p r p
t
r
n
p
p m z m X f
t
z ∑ ∑ ∑
= = =
− +
1
0
1 1
) (
1
 (2.1), 
where z1,...,zn is a solution of the linear sys-
tem of equations: 
() [] q p
n
p
pq pq td z d t c 0
1
1 = − + ∑
=
 ( n q , 1 = ) (2.2) 
or of the equivalent linear system of equa-
tions: 
() q p
n
p
pq pq tb z tb a 0
1
= + ∑
=
 ( n q , 1 = ) (2.3) 
Remark 2.1 Inserting the relations (1.8) and 
(1.9) into (2.2) we observe that (2.2) is 
equivalent to (2.3). Indeed, let   be fixed. 
We have:
q
 
() [] [ () ] ( ∑ ∑ ∑
= = =
⇔ = − + + ⇔ = − +
n
p
pq q p pq pq
n
p
pq q p
n
p
pq pq a tb z b t b a td z d t c
1
0
1
0
1
1 1 + 
+bpq+tbpq ) pq b − zp = tb0q    () q p
n
p
pq pq tb z tb a 0
1
= + ⇔∑
=
Proof of Theorem 2.1: we have to examine 
the solution to the problem: 
() ( )
⎪ ⎭
⎪
⎬
⎫
⎪ ⎩
⎪
⎨
⎧
⎥
⎦
⎤
⎢
⎣
⎡
− − ∑∑
==
2
11
0 0
, 0
n
p
pr r
t
r
p X f E Min α α θ μ
α α
 
(2.4), 
where: ( )
r p pr , α α = . Since (2.4) is the mini-
mum of a positive definite quadratic form, it 
suffices to find a solution with all partial de-
rivatives equal to zero. 
Let: Informatica Economică, nr. 1 (41)/2007  119
f = f( () ( )
⎪ ⎭
⎪
⎬
⎫
⎪ ⎩
⎪
⎨
⎧
⎥
⎦
⎤
⎢
⎣
⎡
− − = = = ∑∑
==
2
11
0 0 0 ) , 1 , , 1 ; ,
n
p
t
r
pr r p pq X f E t r n p α α θ μ α α  = 
=  +  [ () ( ) () () ( ) ∑ ∑ − − ⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
+ +
r p
pr r p
r p
pr r p X f X f E
,
0 0 0
2
,
2
0
2
0 2 2 α θ μ α θ μ α α θ μ
+  = E  -
2
()⎥
⎦
⎤
∑
r p
pr r p X f
,
0 2 α α () [] () ( ) [] θ μ α α α θ μ 0 0
2
,
2
0
2
0 2 E X f E
r p
pr r p −
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
+ + ∑
() ( ) [ ] ( ) [ ] pr
r p
r p pr
r p
r p X f E X f E α α α θ μ ∑ ∑ +
,
0
,
0 2  
Taking the partial derivative with respect to 
0 α  gives the following equation: 
() [] () [ ] 0 2 2 2
,
0 0 = + − ∑ pr r p
r p
X f E E α θ μ α , or: 
() [] () [ ] 0 ,
,
0 0 = + − ∑ r p
r p
r p X f E E α θ μ α  (2.5) 
Using (1.3) in (2.5) we obtain 
 
∑ − =
r p
p prm m
,
0 0 α α  (2.6) 
Inserting this expression for  0 α  into (2.4) 
leads to the following problem: 
() ( ) [] ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎭
⎬
⎫
⎩
⎨
⎧
− − − ∑∑
==
2
11
0 0
n
p
t
r
p r p pr m X f m E Min α θ μ
α
 (2.7) 
Let:  () ( ) [] =
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎭
⎬
⎫
⎩
⎨
⎧
− − − = = = ∑
2
,
0 0 ) , 1 , , 1 ; (
r p
p r p pr pr m X f m E t r n p f α θ μ α  
() { () [] ( ) [ ]⋅ − + − + + = ∑∑ ∑ p r p r p
p pr r
pr p r p
r p
pr m X f m X f m E ' '
'' .,
2
,
2 2
0
2
0 2 α α α θ μ  
( ) [ ] () ( ) ( ) [ ] ∑ ∑ ⋅ + − − − − ⋅
r p
pr p r p
r p
pr p r p m m X f m m X f
,
0
,
0 0 0 2 2 2 ' ' ' α α θ μ θ μ  
() [] } () [ ] ( ) [ ] { } ⋅ + − + + = − ⋅ ∑∑ ∑
' ,' ,
2
,
2 2
0
2
0 2
p pr r
pr p r p
r p
pr p r p m X f E m E m X f α α θ μ  
() [ ] ( ) [ ] { } ( ) [ ] ( ) [ ⋅ − − − − ⋅ ∑ θ μ α θ μ α 0
,
0 0 ' 2 2 ' ' ' ' E E m m X f m X f E
r p
pr p r p p r p r p  
() ( )] ( ) [ ] ( ) [ ] + + + = − + − ⋅ ∑ ... 2
2
0
2
0
,
0 m E m X f E m m X f p r p
r p
pr p r p θ μ α  
() [] { } ( [ ( ) ] { ( ) [ − − + + + − + ∑
≠
≠
' '
'
'
'
' '
, '
2 2 ... 2 ...
r p
r r
p p
r p
p r p r p pr p r p pr X f m X f E m X f E α α α  
]} )( ) [ ] ( ) ( ) ( ) [ ] ( ) ⋅ + + − + − − + − 0 0 0 0 2 ... ... 2 2 ... ' m m X f E E m m p r p pr p θ μ α θ μ  
() ( ) [ ] ( ) ... ... + − + ⋅ p r p pr m X f E α  
On putting the derivatives with respect to  pr α  equal to zero, we get the following system of 
equations: 
() [] { } ( ) [ ] ( ) [ ] { }− − − + − ∑
≠
≠
' ' '
'
'
' '
' '
,
2 2 2
p r p p r p
r r
p p
r p
r p p r p pr m X f m X f E m X f E α α 2 
() ( ) () [] ( ) [] ( ) t r n p m X f E m m X f E p r p p r p , 1 ; , 1 , 0 2 0 0 = = = − + − ⋅ θ μ , that is: Informatica Economică, nr. 1 (41)/2007  120 
() [] () [ ] { } ( ) [ ] ( ) [ ] { } ' ' '
' '
' '
,
0 0 p r p p r p
r p
r p p r p m X f m X f E m X f m E − − = − − ∑α θ μ  (p =  ) , 1 , , 1 t r n = , or: 
() () () [] () () ( ) [ ] { } [ { − = − − ∑ ) (
' '
' '
,
0 0 r p
r p
r p r p r p X f E X f E X f E E α θ μ θ μ E(fp(Xr))] 
() [ () ( )]}( ) t r n p X f E X f
r p r p , 1 , , 1 , ' ' ' ' = = − ⋅  
So: Cov () ( ) [] () ( ) [ ] ' '
' '
' ' , ,
,
0 r p r p
r p
r p r p X f X f Cov X f ∑ = α θ μ    (2.8) 
( ) t r n p , 1 , , 1 = = . Substituting  with q, r with r p
’ in (2.8) one obtains: 
 (2.9)  Cov () () [] ()( [] ∑∑
==
=
n
p
t
r
r p r q r p r q X f X f Cov X f
1 '1 '
' ' ' ' ' 0 ' ' , , α θ μ )
( ) t r n q , 1 ' , , 1 = = . Substituting  ' with   with  p ' ,r p r  in (2.9) one obtains: 
() ( ) [] ()() [] ∑∑
==
=
n
p
t
r
r q r p pr r q X f X f Cov X f Cov
11
' ' 0 , , α θ μ , t r n q , 1 ' , , 1 ( = = )  (2.10) 
From (1.5), (1.7), (1.9) we get 
Cov () ( ) [ ] ( ) () [ ] ( ) ( ) [ ] θ μ θ μ θ μ θ μ 0 0 0 0 0 , , , ' ' q q q r q r q Cov b d X f Cov X f = = = =  =  
=Cov () () [ ] ( ) q q q d b 0 0 0 , = = θ μ θ μ  (2.11), 
where q = t r n , 1 , , 1
' = . Let r, r
’ = t , 1  with r
' r ≠ . From (1.1), (1.5), (1.9) we get 
Cov ()( ) [ ] () ( ) [ ] ( ) [ ] ( ) [ ] ' ' ' ,
r q r p r q r p r q r p X f E X f E X f X f E X f X f − =  =  
= E () ( ) [ ] { } () [ ] { } ( ) [ ] { }= − θ θ θ | | | ' ' r q r p r q r p X f E E X f E E X f X f E  
= E () [ ] ( ) [ ] { } () [ ] ( ) [ ] ( ) ( ) [ ] θ μ θ μ θ μ θ μ θ θ q p q p r q r p E E E X f E X f E = − | | ' - 
-E () [ ] () [ ] () () [ ] pq pq q p q p d b Cov E = = = θ μ θ μ θ μ θ μ ,  (2.12), 
for all p, q =  n , 1 . 
In view of (1.1), (1.4), (1.5), (1.8) and (2.12), we calculate the summation: 
()() [] ()() [] ⋅ + = ∑ ∑
≠
=
'
' ' ' ' , ,
1
r r
r
pr r q r p pr r q r p
t
r
pr X f X f Cov X f X f Cov α α α Cov[fp(Xr), 
,fq( )] =  ' r X ( ) ( ) ( ) [ ] { ( ) ( ) [ , | | , ' ' ' ' θ θ α
r p r q r p pr X f E Cov X f X f Cov E + E(fq(Xr’)| 
| ]}( ) ( ) [ ] { } ∑ ∑
≠ ≠
+ + = +
' '
' , )
r r
r
pr pq q p pq
r r
r
pr pq pr d Cov a d α θ μ θ μ α α θ =  ' pr α (apq+bpq)+ 
+dpq· ∑ ∑
≠ ≠
+ =
'
'
' r r
r
pr pq pq pr
r r
r
pr d c α α α  (2.13), 
where: p,q =  t r n , 1 , , 1
' = . 
Introducing (2.11) and (2.10) in (2.13), one obtains: 
() t r n q d c b
n
p
r r
r
pr pq pq pr q , 1 , , 1 ,
'
1
0
'
' = =
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
+ =∑∑
=
≠
α α  (2.14) 
Because of the symmetry in time clearly: 
p pt p p α α α α = = = = ... 2 1  (2.15) 
and so the system of equations (2.14) can be written as: Informatica Economică, nr. 1 (41)/2007  121
b0q =  , () [] pq pq
n
p
p d t c 1
1
− + ∑
=
α ) , 1 n q =
pq pq
n
p
p d t c 1
1
− + ∑
=
α
( , or (see (1.9)): 
d0q =  , () [] ) , 1 n q = (  (2.16) 
Comparing (2.2) with (2.16), we conclude that: 
t z p p / = α  (p =  n , 1 ) (2.17) 
Remark 2.2 The relation (2.15) can be proven as follows; for r’ = 1, r’ = 2,…., r’ = t-1, r’ = t 
the system of equations (2.14) implies: 
b0q =  () , , 1 ; ...
1
2 1 n q d d c
n
p
pq pt pq p pq p = + + + ∑
=
α α α  
b0q =  () , , 1 ; ...
1
3 1 2 n q d d d c
n
p
pq pt pq p pq p pq p = + + + + ∑
=
α α α α  
b0q =  () , , 1 ; ...
1
2 , 1 1 , n q d d d c
n
p
pq pt pq t p pq p pq t p = + + + + ∑
=
− − α α α α  
b0q =  () n q d d d c
n
p
pq t p pq p pq p pq pt , 1 ; ...
1
1 , 2 1 = + + + + ∑
=
− α α α α . 
The first two equalities imply: 
( ) ( ∑ ∑ + + = + + +
p
pq p pq p pq p
p
pq pt pq p pq p d c d d d c 3 2 1 2 1 ... α α α α α α +…+ 
+ ) pq ptd α , that is: 
 
( ) + + = + + + + ∑ ∑ ∑ ∑ ∑ pq
p
p
pp
pq p pq pt pq p pq
p
p pq
p
p c d d d d c 2 1 3 2 1 ... α α α α α α  
+ () n q d d
p
pq pt pq p , 1 , ... 3 = ∀ + + ∑ α α , or: 
()(), , 1 ,
1
2
1
1 n q d c d c pq pq
n
p
p pq pq
n
p
p = ∀ − = − ∑ ∑
= =
α α  
that is (see (1.8) and (1.9)): 
n q a a
n
p
pq p
n
p
pq p , 1 ,
1
2
1
1 = ∀ =∑ ∑
= =
α α  (2.18) 
Hence, for q = 1 we have: 
ℕ* and from 
here for n=1, we get:
∈ ∀ =∑ ∑
= =
n a a
n
p
p p p
n
p
p ,
1
1 2 1
1
1 α α
, 11 12 11 11 a a α α =  so: 
12 11 α α =  (2.19), 
because  .  0 11 ≠ a
Next, for q = 2 one obtains: 
ℕ* and from 
here for n = 2, we get: 
∈ ∀ =∑ ∑
= =
n a a p
n
p
p p
n
p
p , 2
1
2 2
1
1 α α
22 22 12 12 22 21 12 11 a a a a α α α α + = +  that is (see 
(2.19)):  22 22 22 21 a a α α = , so: 
22 21 α α =  (2.20), 
because  0 22 ≠ a . 
Next, for q = 3 one obtains: 
ℕ* and from 
here for n = 3, we get: 
∈ ∀ =∑ ∑
= =
n a a p
n
p
p p
n
p
p , 3
1
2 3
1
1 α α
33 32 23 22 13 12 33 31 23 21 13 11 a a a a a a α α α α α α + + = + +
, that is (see (2.19), (2.20)):  33 32 33 31 a a α α = , 
so: 
32 31 α α =  (2.21), 
because 0 33 ≠ a . 
From (2.19), (2.20) and (2.21), we may con-
clude that: 
2 1 p p α α = , for all p =  n , 1  (2.22) 
The equality (2.22) can be proven by induc-
tion. Indeed, for n = 1 is readily performed. Informatica Economică, nr. 1 (41)/2007  122 
We assume that: 
1 , 1 , 2 1 − = ∀ = n p p p α α  
From (2.18) for q = n, we have: 
, that is:  pn
n
p
p pn
n
p
p a a ∑ ∑
= =
=
1
2
1
1 α α
n n n n n nn n n n n n n a a a a a a a , 1 2 , 1 2 22 1 12 1 , 1 1 , 1 2 21 1 11 ... ... − − − − + + + = + + + + α α α α α α α
+ 
+ nn n a 2 α and so:  nn n nn n a a 2 1 α α = . From here, 
we get:  2 1 n n α α = , because  . From 
the following two equalities and proceeding 
in the same way, one obtains: 
0 ≠ nn a
3 2 p p α α = , for all p =  n , 1  (2.23) 
Finally, from the last two equalities and pro-
ceeding in the same way, one obtains: 
pt t p α α = −1 , , for all p =  n , 1  (2.24) 
The relations (2.22), (2.23), and (2.24) imply 
(2.15). Now (2.6) and (2.17) lead to: 
∑ p pm z ∑ ∑∑
= ==
− = − = − = p
n
p
p
p
n
p
t
r
p m m
t
z
t m m
t
z
m 0
1
0
11
0 0 α
=
n
p 1
 (2.25) 
Consequently: 
() ( ) () p
n
p
p r p m z m X f ∑
=
− +
1
0
t
r
n
p
p
n
p
pr r
t
r
p t
z X f M ∑ ∑ ∑∑
= = ==
= + = =
1 1 11
0 0
^ 1
α α θ μ
, 
as was to be proven (see (2.4), (2.15), (2.17) 
and (2.25)). 
For the special case when n=1, Theorem 2.1 
reads: 
Theorem 2.2 (optimal non-homogeneous 
linearized estimator, n = 1) 
The linear combination of 1 and the random 
variables f1(Xr) (r =  t , 1 ) closest to  () θ μ0  and 
to f0(Xt+1) in the least squares sense equals: 
() 1 0 1
1
1
zm m X f
t
z M r
t
r
− + = ∑
=
 (2.26), 
where: m1 = E[f1(Xr)], z = td01/{c11+(t-1)d11} 
with: d01 = Cov[f0(Xr), f1(Xr’)], d11 = 
Cov[f1(Xr), f1(Xr’)] for  ' r r ≠ , c11 = 
Cov[f1(Xr), f1(Xr)]. 
Proof: 
For n=1 the relation (1.3) implies: m1 = 
E[f1(Xr)]. For n=1 the relation (1.6) implies: 
c11 = Cov[f1(Xr), f1(Xr)]. For n=1 the linear 
system of equations (2.2) reads: [c11+(t-
1)d11]z = td01 which is equivalent to the fol-
lowing equation:z = td01/{c11+(t-1)d11}. From 
(2.12) for n=1 one obtains: d01 = Cov[f0(Xr), 
f1(Xr’)], d11 = Cov[f1(Xr), f1(Xr’)], where 
' r r ≠ . Finally, for n=1 the relation (2.1) 
reads: 
M = z () 1 0 1
1
1
zm m X f
t
r
t
r
− + ∑
=
()
⎪ ⎭
⎪
⎬
⎫
⎪ ⎩
⎪
⎨
⎧
⎥
⎦
⎤
⎢
⎣
⎡
−∑∑
==
+
2
11
1 0 ) (
n
p
t
r
r pr pr t X f c X f MinE
. 
So the theorem is proven. 
 
Conclusions 
This paper is an original approach of a more 
general credibility model. 
We obtained a semi-linear credibility model, 
which involves the class of linear combina-
tions of given functions of the observable 
variables, for solving the minimization prob-
lems of the type: 
. So 
the approximation to f0(Xt+1) or to 
[ ] ( ) ( ) μ θ | 1 0 0 + = θ t X f E  furnished in this arti-
cle, is based on prescribed approximating 
functions f1, f2,…,fn. The usefulness of this 
approximation is that it is easy to apply, since 
it is sufficient to know estimates for the pa-
rameters apq, bpq appearing in the credibility 
factors zp. The estimators mainly considered 
here are linear functions of several functions 
f1, f2,…, fr of the observable random vari-
ables, which represents claim amounts, rather 
than the claim amounts themselves. For this 
reason, semi-linear credibility estimators, 
which are linear functions of transformed ob-
servations lead to easily computable premi-
ums. This semi-linear credibility results are 
the most recent developments in credibility 
theory and they certainly present the only so-
lution where insurance industry faces risks 
with basic risk characteristics that cannot be 
assigned to any established collective or with 
a risk coverage under circumstances not ear-
lier met. We give a rather explicit description 
of the input data for the model used, only to 
show that in practical situations there will 
always be enough data to apply semi-linear 
credibility theory to a real insurance portfo-
lio. The point we want to emphasize is that 
practical application of semi-linear credibil-Informatica Economică, nr. 1 (41)/2007  123
ity is feasible nowadays using the greatest 
accuracy credibility theory. 
 
References 
[1].  Atanasiu V., Contributions to the 
credibility theory, doctoral dissertation, Uni-
versity of Bucharest-Faculty of Mathematics, 
2000. 
[2]. Atanasiu V., Un model de credibilitate, 
Revista Studii şi Cercetări de Calcul Econo-
mic şi Cibernetică Economică, XXXII, nr.3, 
1998. 
[3].  Goovaerts M.J., Kaas R., Van Heer-
waarden, Bauwelinckx T., Insurance Se-
ries, volume 3, Effective Actuarial Methods, 
University of Amsterdam, The Netherlands, 
1991. 
[4].  Pentikäinen T., Daykin C.D.,Pesonen 
M.,  Practical Risk Theory for Actuaries, 
Université Pierré et Marie Curie, 1990. 
[5]. Sundt B., An Introduction to Non-Life 
Insurance Mathematics, Veröffentlichungen 
des Instituts für Versicherungswissenschaft 
der Universität Mannheim Band 28, VVW 
Karlsruhe, 1990. 

An Introduction to Non-Life Insurance Mathematics, Veröffentlichungen des Instituts für Versicherungswissenschaft der Universität Mannheim Band 28, VVW

Contributions to the credibility theory, doctoral dissertation,

Practical Risk Theory for Actuaries, Université Pierré et Marie Curie,

Un model de credibilitate,

Semi-linear credibility results

http://revistaie.ase.ro/content/41/Semi-Linear%20Credibility%20Results-atanasiu%20virg.pdf

Semi-linear credibility results

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