SHORT CONTRIBUTION
A TEACHER'S REMARK ON EXACT CREDIBILITY
BY HANS U. GERBER
In the classical Bayesian approach to credibility the claims are conditionally
independent and identically distributed random variables, with common density/(x,
ft). The unknown parameter ft is a realization of a random variable © having initial
(prior) density u (ft). Let
H(ft) = \xf(x, ft)dx.
The initial pure premium is
£[X,j = E[ft(9)].
The premium for X / + ! , given X,, ..., X,, is the conditional expectation
9)|X,, ..., X,] = \fi(ft)u(ft\xx,
..., X,)dft.
A central question is for which pairs f(x, ft) and u (ft) this expression is linear, i.e.
of the form
ZX+
(1 -Z)-E
[/*(&)]
where X = (X, + ... + X,)lt is the observed average. This is indeed the case for about
half a dozen famous examples. JEWELL (1974) has found an elegant and general
approach to unify these examples, see also GOOVAERTS and HOOGSTAD (1987,
chapter 2). The classical examples can be retrieved as special cases; however a
preliminary reparameterization has to be performed on a case by case basis. The
purpose of this note is to propose an alternative (but of course strongly related)
formulation of the general model, from which the classical examples can be
retrieved in a straightforward way.
The common density of the claims is supposed to be of the form
a(x)-b(ft)x
f(x, ft) = —-—— ,
c(ft)
x e A.
Here A is the set of possible values of the claims (discrete or continuous), and c (ft)
is the normalizing constant:
-J.,
a(x)-b(ft)xdx.
c(ft) =
)A
JA
ASTIN BULLETIN, 1995, Vol. 25, No. 2, 189-192
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190
HANS U. GERBER
Then
As a prior density we choose
d(n0, x0)
where § varies in some interval,
d(no,xo) =
is the normalizing constant and n0 and x0 are two parameters. Then it is easy to see
that the posterior density
K ( 0 U , , ..., Xt)
is a member of the same family, with updated parameter values:
nt = no + t,
xt = x0 + Xi + ... + X,.
Hence, if we have an expression for E[fi (0)], it suffices to replace « 0 by n, and xn
by x, to obtain E[n(@)\Xu ..., X,].
By definition,
E[fi(0)] = \fi(ff)-u(&)d& =
J
d(n0, x0) J
Now we perform a partial integration and assume that the function
n+l
vanishes at the integration limits.
Then we obtain
V
E[fi(0)] =
=
-4- 1
f
(&)-"»b(§)x°b'(&)d§
I 1
Y
°
nQQ-d(n
-d(n00, , xx00)) J
Hence the premium for Xt + , is
x, + 1
x0 + t • X + 1
n,
n0 +1
no + i
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A TEACHER'S REMARK ON EXACT CREDIBILITY
191
The classical examples can be retrieved directly as follows:
a. POISSON-GAMMA
A = {0, 1, 2, ...}, 0<#<°o
a(x) = —, b(§) = •», c(§) = e§
x\
oa
u(§) = J-—&a-]e-p{>
riot)
with a = xQ+\, p = n0
b . GEOMETRIC-BETA
A = (0, 1, 2, ...}, 0 < # < l
a(x)= 1, b(§)=\
-•&, c(&) = §
u(ff) = — ( —fL§ a - l (i-§f- 1
with a = « 0 +l, P = xo+1
np
C. EXPONENTIAL-GAMMA
A = (0, =0),
a(x)=l,
0a
u(§) = J _ _ ^ « - 1 e - ^ with a = no+\,
r()
d. NORMAL-NORMAL
a(x) - exp - — - L b($) = exp — , c(0) =
M(I?) =
—e V 2«2 / with fi = —
, v2 = —
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192
HANS U. GERBER
C BERNOULLI-BETA
A = {0, 1}, 0 < # < l
a(x)=l,
b{&) =
, c(#) =
# " ' ' ( ! - ! » / " ' with a =
JCQ+1,
fi =
no-xo-\
ACKNOWLEDGMENT
At the ASTIN Colloquium in Leuven the author had fruitful and joyful discussions
with Hans Buhlmann and William Jewell. They contributed to an improvement of
this note.
REFERENCES
GOOVAERTS, M.J. and HOOGSTAD, W. J. (1987) Credibility Theory. Surveys of Actuarial Studies, No. 4.
Nationale-Nederlanden.
JEWELL, W. S. (1974) Credible means are exact for exponential families. ASTIN Bulletin, 8, 77-90.
JEWELL, W.S. (1975) Regularity conditions for exact credibility. ASTIN Bulletin, 8, 336-341.
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