In this article we will obtain, just like in the case of classical credibility model, a credibility solution in the form of a linear combination of the individual estimate (based on the data of a particular state) and the collective estimate (based on aggregate USA data). Mathematics Subject Classification: 62P05.linearized regression credibility premium, the structural parameters, unbiased estimators.

Carmen Lenuta TRICA

English

Research Papers in Economics

Revista Informatica Economică nr.3(47)/2008 
 
133
The Hachemeister Regression Model 
 
Carmen Lenuţa TRICĂ 
Agro-Food and Environmental Economics, Academy of Economic Studies 
Mihail Moxa nr. 7, sector 1, Bucharest, 010961, România 
carmen_trica@yahoo.com 
 
In this article we will obtain, just like in the case of classical credibility model, a cre-
dibility solution in the form of a linear combination of the individual estimate (based on the 
data of a particular state) and the collective estimate (based on aggregate USA data). 
Mathematics Subject Classification: 62P05. 
Keywords: linearized regression credibility premium, the structural parameters, unbiased es-
timators. 
 
ntroduction 
The article contains a description of the 
Hachemeister regression model allowing for 
effects like inflation. 
In Section 1 we give Hachemeister’s original 
model, which involves only one isolated con-
tract. His original model, involving only one 
contract, contains the basics of all further re-
gression credibility models. In this section 
we will give the assumption of the Hache-
meister regression credibility models and the 
optimal linearized regression credibility pre-
mium is derived. 
Section 2 describes the classical Hache-
meister model. In the classical Hachemeister 
model, a portfolio of contracts is studied. Just 
as in Section 1 we will derive the best linea-
rized regression credibility premium for this 
model and we will provide some useful esti-
mators for the structure parameters. 
 
1. The original regression credibility mod-
el of Hachemeister 
In the original regression credibility model of 
Hachemeister, we consider one contract with 
unknown and fixed risk parameterθ , during 
a period of t ( 2 ≥ ) years. The yearly claim 
amounts are denoted by t X X ,..., 1 . Suppose 
t X X ,..., 1  are random variables with finite 
variance. The contract is a random vector 
consisting of a random structure parameter θ  
and observations t X X ,..., 1 . Therefore, the 
contract is equal to( ) ' , X θ , 
where () t X X X ,..., ' 1 = . For this model we 
want to estimate the net premium: 
( ) ( ) θ θ μ | j X E = ,  t j , 1 =  for a contract with 
risk parameter θ . 
Remark 
In the credibility models, the pure net risk 
premium of the contract with risk parameter 
θ  is defined as: 
( ) ( ) t j X E j , 1 , | = ∀ = θ θ μ   (1.1) 
Instead of assuming time independence in the 
net risk premium (1.1) one could assume that 
the conditional expectation of the claims on a 
contract changes in time, as follows: 
( ) ( ) () t j b Y X E
j
j j , 1 , ' |
~ ~ = ∀ = = θ θ θ μ  (1.2) 
where the design vector 
j
Y
~
 is known (
j
Y
~
 is a 
column vector of length q, the non-random (q×1) 
vector 
j
Y
~
 is known) and where the  () θ
~ b  are the 
unknown regression constants ( () θ
~ b  is a column 
vector of length q). 
Remark 
Because of inflation we are not willing to as-
sume that  ( ) θ | j X E  is independent of j. In-
stead we make the regression assump-
tion ( ) ( ) θ θ
~ ~' | b Y X E
j
j = . 
When estimating the vector 
~
β  from the ini-
tial regression hypothesis  ( )
~ ~' β
j
j Y X E =  
formulated by actuary, Hachemeister found 
great differences. He then assumed that to 
each of the states there was related an un-
known random risk parameter θ  containing 
the risk characteristics of the state, and that’s 
from different states were independent and 
I Revista Informatica Economică nr.3(47)/2008 
 
134
identically distributed. Again considering one 
particular state, we assume 
that ( ) () θ θ
~ ~' | b Y X E j j = , with  () [ ]
~ ~ β θ = b E . 
After these motivating introductory remarks, 
we state the model assumptions in more de-
tail. 
Let  ()
'
1 ~ ,..., t X X X =  be an observed random 
() 1 × t  vector and θ  an unknown random risk 
parameter. We assume that:  
(H1)  ( ) () θ θ
~ ~ ~ | b Y X E = . 
It is assumed that the matrices:  
(H2)  () [ ]( )
) (
~ ~ ~ ~ ,
q q b Cov
× Λ = Λ = Λ θ  
(H3)  ( ) [ ]( )
) (
~ ~ ~ ~ , |
t t X Cov E
× Φ = Φ = Φ θ  
are positive definite. We finally introduce:  
(H4)  () [ ]
~ ~ β θ = b E  
Let  j
~
μ  be the credibility estimator of  ( ) θ μ j  
based on
~ X . The optimal choice of  j
~
μ  is de-
termined in the following theorem: 
Theorem 1.1: The credibility estimator  j
~
μ  is 
given by: 
() ⎥ ⎦
⎤
⎢ ⎣
⎡ − + =
~ ~
^
~ ~
'
~
~
β μ Z I b Z Y
j
j  (1.3), 
with: ()
~
1
~
'
~
1
~
1
~ ~
^
~ X Y Y Y b
− − − Φ Φ =    (1.4), 
( )
1
~
1
~
'
~ ~ ~ ~
1
~
'
~ ~ ~
− − − Φ Λ + Φ Λ = Y Y I Y Y Z  (1.5), 
where
~ I  denotes the  q q×  identity matrix 
⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
= =
×
×
) (
~ ~
) 1 ( ^
~
^
~ ;
q q
q
z z b b  for some fixed j. For 
demonstration see [1] from References. 
Remark 
By the credibility estimator of a vector we 
shall mean the vector of the credibility esti-
mator of each element of the vector to be es-
timated. Let 
~
~
μ  be the credibility estimator of 
the vector () () θ θ μ
~ ~ ~
b Y = , where the non-
random  () q t×  matrix 
~ Y  is known. Then we 
clearly have: 
( ) ⎥ ⎦
⎤
⎢ ⎣
⎡ − + =
~ ~ ~
^
~ ~ ~
~
~
β μ Z I b Z Y  (1.6) 
An interesting special case is 
~ ~ I Y = , where 
we get: 
( )
~ ~ ~
^
~ ~
~
~ β Z I b Z b − + =  (1.7) 
as the credibility estimator of  () θ
~ b . 
We introduce the following definition: 
Definition: 
Let  ( ) θ r  be a real-valued function of θ  and 
^
r  an estimator of  ( ) θ r . We shall say that 
^
r  
is a θ -unbiased estimator of () θ r , if 
() θ θ r r E = ⎟
⎠
⎞
⎜
⎝
⎛ |
^
 a.s.. 
By the requirement of θ -unbiasedness 
for ( ) θ μ j , we obtain the following theorem: 
Theorem 1.2: The best linear θ -unbiased es-
timator of  ( ) θ μ j  based on 
~ X  is:  
^
~
'
~
~
b Y
j
j = μ  (1.8) 
for some fixed  t j , 1 = . For demonstration see 
[1] from References. 
Remark 
As by the credibility estimators, we directly 
transfer the results of Theorem 1.2, to estima-
tion of vectors and get that 
^
~ ~
^
~
b Y = μ  is the 
best linear θ -unbiased estimator of 
( ) ( ) θ θ μ
~ ~ ~
b Y =  and as a special case, that 
^
~ b  is 
the best linear θ -unbiased estimator of  () θ
~ b . 
This last result gives the following interpreta-
tion of (1.7). 
Theorem 1.3: The credibility estimator of 
( ) θ
~ b  (so 
~
~
b ) is a weighted mean of the best 
linear  θ -unbiased estimator of  () θ
~ b  (so 
^
~ b ) 
and the expectation of  ( ) θ
~ b  (so 
~
β ). 
2. The classical credibility regression mod-
el of Hachemeister 
In this section we will introduce the classical 
regression credibility model of Hachemeister, Revista Informatica Economică nr.3(47)/2008 
 
135
which consists of a portfolio of k contracts, 
satisfying the constraints of the original Ha-
chemeister model. 
The contract indexed j is a random vector 
consisting of a random structure  j θ  and ob-
servations jt j X X ,..., 1 . Therefore the contract 
indexed j is equal to ( )
' , j j X θ , where 
() jt j j X X X ,..., 1
' =  and  k j , 1 =  (the variables 
describing the 
th j  contract are 
( ) k j X j j , 1 , ,
' = θ ). Just as in Section 1 we 
will derive the best linearized regression cre-
dibility estimators for this model. 
Instead of assuming time independence in the 
net risk premium: 
( ) ( ) t q k j X E j jq j , 1 , , 1 , | = = = θ θ μ  (2.1) 
one could assume that the conditional expec-
tation of the claims on a contract changes in 
time, as follows: 
() ( ) ( ) t q k j y X E j jq j jq j q , 1 , , 1 , | = = = = θ β θ θ μ  (2.2), 
with  jq y  assumed to be known and  () ⋅ β  as-
sumed to be unknown. 
Consequence of the hypothesis (2.2): 
() () ( ) k j x X E j
n n t
j j j
t , 1 , |
) 1 , ( ) , ( ) 1 , ( = = = θ β θ θ μ  (2.3), 
where 
) , ( n t x  is a matrix given in advance, the 
so-called design matrix, and where the  ( ) j θ β  
are the unknown regression constants. Again 
one assumes that for each contract the risk 
parameters  ( ) j θ β  are the same functions of 
different realizations of the structure parame-
ter. 
For some fixed design matrix 
) , ( n t x  of full 
rank n ( t n < ), and a fixed weight matrix 
) , ( t t
j v , the hypotheses of the Hachemeister 
model are: 
(H1) The contracts ( )
' , j j X θ  are independent; 
the variables  k θ θ ,..., 1  are independent and 
identically distributed. 
(H2)  ( ) () k j x X E j
n n t
j
t
j , 1 , |
) 1 , ( ) , ( ) 1 , ( = = θ β θ , 
where β  is an unknown regression vector; 
( ) ( )
) , ( 2 ) 1 , ( |
t t
j j j
t
j v X Cov θ σ θ = , where: 
( ) ( ) j jr j X Var θ θ σ |
2 = , t r , 1 = ∀  and 
) , ( t t
j j v v =  is a known non-random weight 
() t t×  matrix, with rg k j t v j , 1 , = = . 
We introduce the structural parameters, 
which are natural extensions of those in the 
Bühlmann-Straub model. We have: 
( ) [ ]
() []
() [] j
n
j
n n
j
E b b
Cov a a
E s
θ β
θ β
θ σ
= =
= =
=
) 1 , (
) , (
2 2
 
), 6 . 2 (
) 5 . 2 (
) 4 . 2 (
 
where  k j , 1 = . 
After the credibility result based on these 
structural parameters is obtained, one has to 
construct estimates for these parameters. 
Write: 
()
()
1 1 ' ) , (
) , (
− − = =
= =
x v x u u
X Cov c c
j
n n
j j
j
t t
j j
 
( )
1 2 ) , ( −
+ = = j
n n
j j u s a a z z = [the resulting 
credibility factor for contract j],  k j , 1 = . 
We can now derive the regression credibility 
results for the estimates of the parameters in 
the linear model. Multiplying this vector of 
the estimates by the design matrix provides 
us with the credibility estimate for  ( ) j θ μ , see 
(2.3). 
Theorem 2.1: (Linearized regression credibil-
ity premium) 
The best linearized estimate of 
( ) [ ] j j
n X E |
) 1 , ( θ β  is given by: 
()
) 1 , ( ) , ( ) , ( ) , ( n n n
j
n n
j
n n
j j b z I B z M − + =  (2.7) 
and the best linearized estimate of 
( ) [ ] j j
n n t X x E |
) 1 , ( ) , ( θ β  is given by: 
( ) [ ]
) 1 , ( ) , ( ) , ( ) , ( ) , ( ) , ( n n n
j
n n
j
n n
j
n t
j
n t b z I B z x M x − + =   (2.8), 
where  j B  is the classical result for the re- gression vector, namely the GLS–estimator Revista Informatica Economică nr.3(47)/2008 
 
136
for  ( ) j θ β  [the vector  j B  minimizing the 
weighted distance to the observations 
() () () j j j j j j j B x X v B x X B d X − − =
−1 ' ,,  
reads 
() ( ) j j j j j j j j j j X c x x c x X v x u X v x x v x B
1 ' 1 1 ' 1 ' 1 ' 1 1 ' − − − − − − − = = =
, in case 
' 2 xax v s c j j + = ], k j , 1 = . 
For proof see [1] from References. 
Remark 
From (2.7) we see that the credibility esti-
mates for the parameters of the linear model 
are given as the matrix version of a convex 
mixture of the classical regression result  j B  
and the collective result b . 
Theorem 2.1 concerns a special contract j. By 
the assumption, the structural parameters  b a,  
and 
2 s  do not depend on j. So if there are 
more contracts, these parameters can be es-
timated. 
Every vector j B  gives an unbiased estimator 
of  b . Consequently, so does every linear 
combination of type  j
j
j B ∑α , where the 
vector of matrices ( )
k j
n n
j , 1
) , (
= α , is such that: 
) , (
1
) , ( n n
k
j
n n
j I = ∑
=
α  (2.9) 
The optimal choice of 
) , ( n n
j α  is determined in 
the following theorem: 
Theorem 2.2: (Estimation of the parameters 
b  in the regression credibility model) 
The optimal solution to the problem 
( ) α
α
d Min  (2.10), where: 
()
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
− ⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
− = − = ∑ ∑ ∑
j
j j
j
j j
def
P j
j j B b P B b E B b d α α α α
'
.
2
= (the distance from 
⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
∑
j
j j B α  to the parameters b ),
 
) , ( n n P P =  a given positive definite matrix 
(P  is a non-negative definite matrix), with 
the vector of matrices  ( )
k j j , 1 = = α α  satisfying 
(2.9), is: 
j
k
j
j
n
B z Z b ∑
=
− =
1
1
) 1 , ( ^
 (2.11), 
where  ∑
=
=
k
j
j z Z
1
 and  j z  is defined as: 
k j u s a a z j j , 1 , ) (
1 2 = + =
−  
For the proof see [1] the chapter 8 from Ref-
erences. 
Theorem 2.3: (Unbiased estimator for 
2 s  foe 
each contract group) 
In case the number of observations  j t  in the 
th j  contract is larger than the number of re-
gression constants n, the following is an un-
biased estimator of 
2 s : 
() () j j j j j j j
j
j B x X v B x X
n t
s − −
−
=
−1 '
^
2 1
(2.12) 
For the proof see [1]., the chapter 8, from 
References. 
Corollary: (Unbiased estimator for 
2 s  in the 
regression model) 
Let K denote the number of contracts j, with 
n t j > . Then 
2
^
2 s s E = ⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
, if: 
∑
>
=
n t j
j
j
s
K
s
;
^
2
^
2 1
 (2.13) 
For a, we give an unbiased pseudo-estimator, 
defined in terms of itself, so it can only be 
computed iteratively: 
 
Theorem 2.4: (Pseudo-estimator for a) 
The following random variable has expected 
value a: 
' ^ ^ ^
1
1
⎟
⎠
⎞
⎜
⎝
⎛ − ⎟
⎠
⎞
⎜
⎝
⎛ −
−
= ∑ b B b B z
k
a j j
j
j (2.14) 
For the proof see [1]., the chapter 8, from 
References. 
Remark: 
Another unbiased estimator for a is the fol-
lowing: Revista Informatica Economică nr.3(47)/2008 
 
137
()() () ()
⎭
⎬
⎫
⎩
⎨
⎧
− − − − − = ∑∑ ∑
= j i
k
j
j j j j i j i j i j u w w w s B B B B w w w w a
,1
.
^
2 ' 2 2
.
^
2
1
/ 1 , (2.15) 
where  j w  is the volume of the risk for the 
th j  contract,  k j , 1 =  and  ∑
=
=
k
j
j w w
1
. . 
Observation: 
This estimator is a statistic; it is not a pseudo-
estimator. Still, the reason to prefer (2.14) is 
that this estimator can easily be generalized 
to multi-level hierarchical models. In any 
case, the unbiasedness of the credibility pre-
mium disappears even if one takes (2.15) to 
estimate a. 
 
Conclusions 
The article contains a credibility solution in 
the form of a linear combination of the indi-
vidual estimate (based on the data of a par-
ticular state) and the collective estimate 
(based on aggregate USA data). This idea is 
worked out in regression credibility theory. 
In case there is an increase (for instance by 
inflation) of the results on a portfolio, the risk 
premium could be considered to be a linear 
function in time of the type  ( ) ( ) θ β θ β 1 0 t + . 
Then two parameters  () θ β0  and  ( ) θ β1  must 
be estimated from the observed variables. 
This kind of problem is named regression 
credibility. This model arises in cases where 
the risk premium depends on time, e.g. by in-
flation. The one could assume a linear effect 
on the risk premium as an approximation to 
the real growth, as is also the case in time se-
ries analysis. 
These regression models can be generalized 
to get credibility models for general regres-
sion models, where the risk is characterized 
by outcomes of other related variables. 
This paper contains a description of the Ha-
chemeister regression model allowing for ef-
fects like inflation. If there is an effect of in-
flation, it is contained in the claim figures, so 
one should use estimates based on these fig-
ures instead of external data. This can be 
done using Hachemeister’s regression model. 
In this article the regression credibility result 
for the estimate of the parameters in the li-
near model is derived. After the credibility 
result based on the structural parameters is 
obtained, one has to construct estimates for 
these parameters. 
The mathematical theory provides the means 
to calculate useful estimators for the structure 
parameters. 
The property of unbiasedness of these esti-
mators is very appealing and very attractive 
from point of view practical. 
The fact that it is based on complicated ma-
thematics, involving conditional expectations 
and conditional covariances, needs not bother 
the user more than it does when he applies 
statistical tools like discriminant analysis, 
scoring models, SAS and GLIM. 
 
References 
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dibility theory, Doctoral Dissertation, Uni-
versity of Bucharest-Faculty of Mathematics, 
2000. 
[2] Daykin, C.D., Pentikäinen, T., Pesonen, 
M.,  Practical Risk Theory for Actuaries, 
Chapman & Hall, 1993. 
[3] De Vylder, F., Optimal semilinear cre-
dibility, Mitteilungen der VSVM, 76, 17-40, 
1976. 
[4] De Vylder, F. & Goovaerts, M.J., Semili-
near credibility with several approximat-
ing functions, Insurance: Mathematics and 
Economics, 4, 155-162, 1985. (Zbl.No. 0167-
6687). 
[5]  Gerber, H.U.,  Credibility for Esscher 
premiums, Mitteilungen der VSVM, 80, 3, 
307-312, 1980. 
[6]  Goovaerts, M.J., Kaas, R., Van Heer-
waarden, A.E.,  Bauwelinckx, T.,  Effective 
Actuarial Methods, vol. 3, Elsevier Science 
Publishers B.V., 187-211, 1990. 
[7] Hogg, R.V. & Klugman, S.A., Loss dis-
tributions, John Wiley and Sons, New York, 
1984. 
[8] Sundt, B., An Introduction to Non-Life 
Insurance Mathematics, volume of the 
“Mannheim Series”, 22-54, 1984. 
 

The Hachemeister Regression Model

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