On credibility evaluation and the tail area of the
exponential dispersion family
Zinoviy Landsman
∗, Udi E. Makov
Department of Statistics, University of Haifa, Mount Carmel, 31905 Haifa, Israel
Received 1 July 1999; received in revised form 1 May 2000; accepted 30 June 2000
Abstract
It has recently been established that the credibility formula for claim distributions belonging to the exponential dispersion family with a known dispersion parameterλprovides the exact predicted mean for a future claim. This paper addresses the case whenλis unknown and when its prior distribution is unspecified. An “optimal” credibility formula is derived for the case when one can specify the probability that claims exceed a given threshold, corresponding to the likelihood of very large claims. © 2000 Elsevier Science B.V. All rights reserved.
Keywords: Fair premium; Credibility formula; Exponential dispersion family; Optimal credibility factor; Tail probability
1. Introduction
Letθ be a risk parameter characterizing a member of a risk collective, and givenθ, letPθ, the distribution of
claimX, be a member of a family of distributions{Pθ, θ ∈Θ⊂R1}. Furthermore, letπ(θ )be the prior distribution
ofθ, the so-called structure distribution.
The estimation of the fair premiumµ(θ )=E(X|θ ), givennyears individual experiencex1, x2, . . . , xnand the
collective fair premiumm=RΘµ(θ )dπ(θ ), is traditionally done by means of a credibility formula of the type
(1−zn)m+znx.¯ (1)
Clearly, the choice of the credibility factorzn is of an immense importance and has attracted a lot of research
interests. In the 1920 American actuaries suggested zn = n/(n+N ), where N was experimentally chosen
(Longley-Cook, 1962). Bu˝hlmann (1967) showed that a minimum least-squares unbiased estimator is obtained forN =Eπ[V (X|θ )]/Vπ[E(X|θ )], whereV (X|θ )is the variance ofXfor givenθ, EπandVπrepresent the mean
and the variance with respect toπ(θ ). Bailey (1950) and Mayerson (1964) noticed that this credibility formula is the exact Bayesian result for particular combinations of prior and claim distributions. The observations of these two authors were generalized by Jewell (1974) who showed that credible means are exact Bayesian for claims whose distribution is a member of the natural exponential family (NEF)
dPθ =exθ−kθdQ(x), θ∈Θ⊂R1, (2)
∗Corresponding author. Tel.:+972-4-824-9003; fax:+972-4-825-3849. E-mail address: landsman@stat.haifa.ac.il (Z. Landsman).
Θ= {θ|k(θ )=lnR eθ xdQ(x) <∞}, and where a conjugate prior distribution is given by
dπ(θ )∝en0(x0θ−k(θ ))dθ. (3)
Consequently, the Bayesian credibility formula takes the form
E(xn+1|x1, . . . , xn)=E(µ(θ )|x1, . . . , xn)=
See also Diaconis and Ylvisaker (1979), Schmidt (1980), Goel (1982), Herzog (1990) and Gerber (1995). For reviews of credibility theory see Goovaerts and Hoogstad (1987) and Herzog (1994). For reviews of Bayesian credibility see Klugman (1992) and Makov et al. (1996).
Landsman and Makov (1998, 1999) discussed credibility in the wider context of the exponential dispersion family (EDF)
dPθ,λ=eλ(xθ−k(θ ))dQλ(x), θ ∈Θ⊂R1, λ∈R+, (5)
which was considered in Tweedie (1984), Nelder and Wedderburn (1972) and Jorgensen (1986, 1987, 1992, 1997). We now summarize the results of Landsman and Makov (1998, 1999) where credibility formulae were derived for the EDF under various assumptions concerning the value of the dispersion parameterλ.
LetEθ,λ(·), Eλ(·)andE(·)denote expectations with respect to the measures dPθ,λ,dπ(θ )dPθ,λandg(λ)dλdπ
(θ )dPθ,λ, respectively. We start by considering the case whereλis known. For EDF (5) let (3) be the conjugate
prior forθ. Then for givenλ∈R+, the credibility formula is given by
The credibility formula (6) is conditional onλ. Ifλis unknown, as is typically the case, one can specify a prior distribution forλwhich we denote byg(λ), and then
E(xn+1|x1, . . . , xn)=m
for anyg(λ), assuming the convergence of both integrals in (7).
Further it was established that the m.s.e. of the predicted mean claim in (7) and the Fisher information functional about scale parameter
are related through an inequality which indicates that a reduction of the m.s.e., up to O(1/n), is achieved by minimizingI (g).
The credibility factor that corresponds to the prior, which minimizes Fisher informationI (g)subject specified restrictions was called the optimal credibility factor. Landsman and Makov (1999) noticed that under natural and wide restrictions on the priorgthe minimum ofI (g)is equal to 1 and this minimum is attained by the class of exponential distributions
{gγ|gγ(λ)=γe−γ λI[λ≥0], γ ∈R+}. (8)
The choice of a particular optimal distribution in (8) can be made possible when the mean ofλis pre-specified
λ0= Z ∞
0
λg(λ)dλ, (9)
In this paper we choose to restrictg(λ)not through (9) but through the specification of the tail area of the claim distribution which corresponds to the likelihood of extremely large claims.
2. Credibility factor and tail area
Suppose a priori an actuary assesses that the probability that a claim exceeds a thresholdT is equal to some small probabilityα, i.e.
for any (θ, λ)∈Θ×R+. Then there are several possibilities to chooseγ and consequently the distributiongγ. One
way consists of averaging (10) with respect togγ and findingγ (θ )as a solution of the following equation:
γ Z ∞
0
¯
F (T , θ, λ)e−γ λdλ=α. (11)
The final value ofγ, sayγ#, is obtained by taking the expectation ofγ (θ )with respect to the conjugate prior ofθ (3),
γ#= Z
Θ
γ (θ )dπ(θ ). (12)
Let us see how this approach can be realized for the case of normally distributed claims.
Theorem 1. Let X be a normal claim size and suppose that one can state some small probabilityα∈(0,1)that X exceeds a large thresholdT >0, i.e.
P (X > T )=α.
Then the credibility factor in the credibility formula
E(xn+1|x1, . . . , xn)=(1−zn)m+znx¯
can be calculated as follows
zn=1−
is the incomplete gamma function (see, for instance, Luke, 1975, Chapter 4).
Proof. LetΦ(x)be standard normal distribution function. Then we can rewrite (11) as follows
γ Z ∞
0
¯
whereµ=θis the expectation of normal claim size. After the integration by the parts of left-hand side of (16) one
Notice that the integrand in the last equation is the kernel of a Gamma distribution with(((T −µ)2/2)
+γ )−1and 1
2being, respectively, the scale and shape parameters. Thus one obtains the equation forγ (µ)
1
Solving this equation we get
γ (µ)= (T −µ)
Taking expectation ofγ (µ)with respect to conjugate prior (3), having in our case the form
dπ(µ)=
Statement (14) is now proved. Statement (13) follows from Landsman and Makov (1999, Theorem 4), substituting
ν=1.
For other members of the EDF the approach summarized in (10)–(12) meets serious analytical difficulties, however it can always be realized if not analytically then numerically. The following theorem suggests using a normal approximation for claim distributions with largeλ.
Theorem 2. Let the distribution of X, the claim size, belong to the EDF with meanµ=µ(θ )=k′(θ )and variance functionV (µ) = V (µ(θ ))= k′′(θ ). Suppose that suppg(λ) = [λ1,∞)and letλ1 be so large that the normal approximation for the distribution of X stated in Jorgensen (1997, Theorem 3.4) is valid. Then the approximately optimal credibility factor can be given by (13) with
γ#=2α(1−α)
Proof. According to Jorgensen (1997, Theorem 3.4) for λ ≥ λ1 when λ1 is large enough, the distribution of Z=√λ(X−µ)/√V (µ)is approximately standard normal. Then changingF (T , θ, λ)¯ byΦ(¯ √λ(T−µ)/√V (µ)) in (16) we obtain, repeating the arguments given in the proof of Theorem 1,
and the stated value ofγ#follows immediately. Notice that the class of “optimal” priors forλis the class ofλ1-shifted
exponential distributions and then the optimal credibility factor has the same expression (13).
Example 1. Normal claims,V (µ)=1,
K(T , P , π )=(T−m)2+ 1 n0
,
and (18) coincides with (14).
Example 2. Gamma distributed claims. For this case
k(θ )= −ln(−θ ), k′(θ )=−1 θ , k
′′(θ )= 1
θ2, θ∈Θ =R−, (20)
and the conjugate distribution ofθgiven in (3) is “negative” Gamma in the sense that distribution of−θis Gamma with shape and scale parameters equaled to(n0+1)and 1/(x0n0), respectively. Then denoting byπ (u)¯ =1− π(−u), u >0 and substituting (20) into (19) we have
K(T , P , π ) = Z ∞
0
(Tu−1)2dπ (u)¯ =T2 n0+1 (n0x0)2+
n 0+1 n0x0 −
1 T
2!
=T2 n0+1 (n0m)2 +
1
m− 1 T +
1 n0m
2!
becausex0=m.
3. Discussions
Seeking to minimize the m.s.e. of the estimated fair premium, Landsman and Makov (1999) found an optimal credibility factor, using class (6) of least favorable distributions of dispersion parameter λ and specifying its expectationλ0=Eλ. In this paper we show how to choose an optimal credibility factor specifying the probability αthat claim size exceeds some thresholdT. The optimal factor for the normal claims is explicitly given in Theorem 1. In Fig. 1 we show how the optimal credibility factors vary withnfor three probability levelsα1 =0.01 (solid
line style),α2=0.05 (dash line style) andα3=0.1 (dots line style) withm=2, T =10, n0=0.75 given in some
scale, say of $1000’s. Here we can see that the weight of the individual experience (x)¯ in the credibility formula decreases if the probability of exceeding the thresholdT increases.
Fig. 2.znfor normal and gamma claims.
This can be explained by the fact that as the value ofαincreases, the tail becomes heavier and the variance of the claims larger. Consequently, the information aboutµ(Fisher information) provided byx¯decreases (see Landsman, 1996, Corollary 1) and so the credibility factor associate withx¯decreases.
Theorem 2 provides an approximate credibility factor for members of the EDF. In Fig. 2 we compare the change of the credibility factorzninnfor the normal (solid line style) and gamma (dots line style) claims, whereα=0.05.
We can see that the weight given to the individual experience in the credibility formula for gamma claims is less than that for the normal claims, corresponding to the same tail probability (i.e.αandT are the same but the tails’ behavior of these two distributions are different). This phenomenon can be explained as follows: for normal claimsx¯, the individual experience, is a sufficient statistics forµ, and consequently provides the maximum amount of information, which is equal the information contained in the whole sample (x1, . . . , xn). For gamma claimsx¯is
not a sufficient statistics of the meanµ, and so it provides a reduced amount of information. This is reflected in the smaller credibility factor associated with gamma claims.
Acknowledgements
We are grateful to an anonymous Referee for his useful comments.
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