An application of nonparametric regression
estimation in credibility theory
q
Weimin Qian
a,b,∗aDepartment of Mathematics and Computer Science, University of Marburg, 35032 Marburg, Germany bDepartment of Applied Mathematics, Tongji University, 200092 Shanghai, PR China
Received January 2000; received in revised form February 2000; accepted February 2000
Abstract
In this paper, we use the nonparametric regression method to establish estimators for credibility premiums under some principles of premium calculation. The asymptotic properties of the estimators are studied. © 2000 Elsevier Science B.V. All rights reserved.
Keywords: Credibility premium; Nonparametric regression; Kernel estimator; Local average estimator
1. Introduction
A principle of premium calculation is a rule, say H. For any risk S, such a principle enables the insurer to quote a premium P=H[S]. This means that the insurer is willing to receive P and in return make a random payment of S.
Thus, the insurer’s gain from such a contract is P−S and is a random variable.
Some important principles of premium calculation are stated, e.g., in Gerber (1979). 1. The net premium principle(principle of equivalence). This means that P=E(S).
2. The expected value principle. This means that there is a safety loading proportional to E(S). Thus P=(1+λ)E(S), whereλ> 0 is a parameter.
3. The variance principle. A safety loading proportional to the variance, i.e. P=E(S)+αVar(S), whereα> 0 is a parameter.
4. The standard deviation principle. A safety loading proportional to the standard deviation, i.e.P = E(S)+ β√Var(S), whereβ> 0 is a parameter.
5. The percentile principle(chance constrained premium). Let 0< ε <1. Then P is determined such that the prob-ability for a loss from the contract is at mostε, i.e.
P =min{p|F (p)≥1−ε}, where F (p)≡P{S≤p}.
Let S1, S2, . . . denote the claims from subsequent periods. Given S1, S2, . . ., Sn, one has to determine an appropriate premium to cover the risk Sn+1. Let us denote Pi as the credibility premium to cover Si.
q
Research supported by an Allianz scholarship of Alexander von Humboldt Foundation.
∗Tel.:+49-6421-2825484.
E-mail address: qianweimin@hotmail.com (W. Qian).
We suppose that S1, S2,. . . is a stationary Markov chain of order q taking its values in E⊂R. According to a
theorem in Bühlmann (1970, p. 97), under mild assumptions we get
Pn+1=H (Sn+1|S1, S2, . . . , Sn).
We find the following equations by using various principles. 1. Net premium principle:
Pn(1+)1=E(Sn+1|S1, . . . , Sn)=E(Sn+1|Sn−q+1, . . . , Sn). 2. Expected value principle:
Pn(2+)1=(1+λ)E(Sn+1|Sn−q+1, . . . , Sn). 3. Variance principle:
Pn(3+)1=E(Sn+1|Sn−q+1, . . . , Sn)+αVar(Sn+1|Sn−q+1, . . . , Sn) =E(Sn+1|Sn−q+1, . . . , Sn)+α(E(Sn2+1|Sn−q+1, . . . , Sn)
−(E(Sn+1|Sn−q+1, . . . , Sn))2). 4. Standard deviation principle:
Pn(4+)1=E(Sn+1|Sn−q+1, . . . , Sn)+β(E(Sn2+1|Sn−q+1, . . . , Sn)−(E(Sn+1|Sn−q+1, . . . , Sn))2)1/2. 5. Percentile principle:
Pn(5+)1=min{p|Fn+1(p)≥1−ε}.
whereFn+1(x)≡P{Sn+1≤x|Sn−q+1, . . . , Sn}.
In this paper we use the nonparametric regression method to establish estimators forPn(i)+1, i=1, . . . ,5. Kernel estimators forPn(i)+1, i =1, . . . ,4 are discussed in Section 2. In Section 3 we improve kernel estimators and study the asymptotic properties of the estimators under some weak moment conditions. We also consider in Section 4 the local average estimator forPn(5+)1introduced by Truong and Stone and examine its asymptotic property.
2. Kernel estimators of the credibility premium
First we use a method proposed by Collomb (1984) to estimate E(Sn+1|Sn−q+1, . . . , Sn) and E(Sn2+1| Sn−q+1, . . . , Sn).
Denote
Rn(u)= Pn−1
i=qSi+1K((u−(Si−q+1, . . . , Si)T)/ hn) Pn−1
i=qK((u−(Si−q+1, . . . , Si)T)/ hn)
, u∈Eq, (2.1)
Rn∗(u)=
Pn−1
i=qSi2+1K((u−(Si−q+1, . . . , Si)T)/ hn) Pn−1
i=qK((u−(Si−q+1, . . . , Si)T)/ hn)
, u∈Eq, (2.2)
We establish the estimator of Pn+1using the following principles.
1. Net premium principle:
ˆ
Pn+1=Rn(Sn−q+1, . . . , Sn). 2. Expected value principle:
ˆ
Pn+1=(1+λ)Rn(Sn−q+1, . . . , Sn). 3. Variance principle:
ˆ
Pn+1=Rn(Sn−q+1, . . . , Sn)+α j
R∗n(Sn−q+1, . . . , Sn)−(Rn(Sn−q+1, . . . , Sn))2 k
.
4. Standard deviation principle:
ˆ
Pn+1=Rn(Sn−q+1, . . . , Sn)+β(R∗n(Sn−q+1, . . . , Sn)−(Rn(Sn−q+1, . . . , Sn))2)1/2.
In order to prove some results about the estimators, we recall Doeblin’s condition, which can be found e.g. in Doob (1953, p. 192).
Assume that (Sn)N is a stationary Markov process of order q, and denote by (,A) its state space. For x∈ and for A∈A, denote by Pm(x, A) the m-step transition probability function of (Sn)N. The process (Sn)Nis said to satisfy Doeblin’s condition if there exists a finite measureψdefined onAwithψ() > 0, a positive integer m and someη> 0 such that for each A∈Awithψ(A)< ηthere holds
Pm(x, A)≤1−η for every x∈. In short, we write
Xi ≡(Si, . . . , Si+q−1)T, Yi ≡Si+q, (2.3)
and introduce some assumptions on (Xi, Yi) and the kernel function K.
Let us denote a compact subset of Rqby G and a compactε-neighborhood ofG(⊂ ˆG)byG:ˆ (A.1) There are positive constantsŴ,νsuch that
P (Xi ∈B)≤Ŵλ(B) for every i∈N and all B∈B(Rq), P (Xi ∈B)≥νλ(B) for every i∈N and all B∈B(G),ˆ
where B(Rq) (resp.B(G)) is theˆ σ-algebra of the Borel sets on Rq (resp. onG) andˆ λthe Lebesgue measure on Rq.
(A.2) There is a positive constant M such that
E(|Yi|β)≤M for some β >2 and for every i∈N. (A.3) There is a positive constant V such that
E((Yi−R(Xi))2|Xi =x)≤V for every i∈N and x∈ ˆG, where R(Xi)=E(Yi|Xi).
(K.1) There is a positive constantK¯ such that
|K(x)| ≤ ¯K <∞ for every x ∈Rq. (K.2)
(K.3) There exists a positive constantKˆ such that
0<
Z
Rq
K(u)du
≤
Z
Rq|
K(u)|du≤ ˆK <∞.
(K.4) K is Lipschitz continuous of orderγon Rq, 0< γ <1.
Theorem 1. Let S(n)N attain its values in a compact setG˜ of R and satisfy Doeblin’s condition. Assume that (A.1)–(A.3) are satisfied by the variables (Xi, Yi) defined as in (2.3) and that the kernel function satisfies (K.1)–(K.4).
Suppose that the bandwidth hnsatisfies
nhqn
log2n → ∞, n→ ∞.
Then we have
| ˆPn(1+)1−Pn(1+)1| →0 as n→ ∞
and
| ˆPn(2+)1−Pn(+2)1| →0 as n→ ∞.
To study the properties ofPˆn(3+)1andPˆn(+4)1we need a further condition. (A.3)∗There is a positive constantV∗such that
E((Yi2−R∗(Xi))2|Xi =x)≤V∗ for every i∈N and x∈ ˆG, whereR∗(Xi)=E(Yi2|Xi).
Theorem 2. Assume that the conditions stated in Theorem 1 hold. If (A.3)∗ holds, (A.2) holds for β> 4 and supx∈G|R(x)|<∞,where R(x)=E(Yi|Xi=x), then
| ˆPn(3+)1−Pn(+3)1| →0 as n→ ∞. (2.4)
Moreover, if infx∈GE(Yi2|Xi =x) > supx∈GR(x) 2
,then
| ˆPn(4+)1−Pn(+4)1| →0 as n→ ∞. (2.5)
Theorem 1 is a corollary of Theorem 3 in Collomb (1984) (see also Theorem 3.4.8 in Györfi et al. (1989)).
Proof. Since
| ˆPn(3+)1−Pn(3+)1| = |Rn(Xn−q+1)+α[Rn∗(Xn−q+1)−(Rn(Xn−q+1))2]−R(Xn−q+1)−α[R∗(Xn−q+1) −(R(Xn−q+1))2]| ≤ |Rn(Xn−q+1)−R(Xn−q+1)| +α|R∗n(Xn−q+1)−R∗(Xn−q+1)| +α|(Rn(Xn−q+1))2−(R(Xn−q+1))2| ≡I1+αI2+αI3. (2.6)
It follows from Theorem 3.4.8 in Györfi et al. (1989) that
I1→0 as n→ ∞, (2.7)
I2→0 as n→ ∞. (2.8)
I3=I1|Rn(Xn−q+1)+R(Xn−q+1)|
for large values of n we find
Rn∗(Xn−q+1)−(Rn(Xn−q+1))2
3. Improved kernel estimators of the credibility premium
In this section we study improved kernel estimators ofPn(i)+1, i=1, . . . ,4.
Let bn, n∈N, be a sequence of positive numbers and hn, n∈N, a sequence of bandwidths. Consider the estimators
We establish the improved kernel estimators of Pn+1in the case of the following principles.
1. Net premium principle:
˜
Pn(1+)1= ˜Rn(Xn−q+1).
2. Expected value principle:
˜
Pn(2+)1=(1+λ)R˜n(Xn−q+1).
3. Variance principle:
˜
Pn(3+)1= ˜Rn(Xn−q+1)+α(R˜n∗(Xn−q+1)−(R˜n(Xn−q+1))2).
4. Standard deviation principle:
˜
Pn(4+)1= ˜Rn(Xn−q+1)+β(R˜n∗(Xn−q+1)−(R˜n(Xn−q+1))2)1/2.
Here we use the notations of Section 2. Moreover, we state some new assumptions.
(A.1)′Assume the existence of a common marginal density f0of the random variables xi. f0is continuous onG,ˆ
and there is a positive constant m1such that
f0(x)≥m1>0 for every x∈G.
(A.2)′There exists a positive constant M such that
E(|Yi|1+δ)≤M for some δ >0 and all i∈N. (A.3)′There exists a positive constant V such that
E(|Yi|1+δ|Xi =x)≤V for some δ >0 and all x∈ ˆG, i∈N.
Theorem 3. Suppose that the variables (Xi, Yi) defined as in (2.3) obey Conditions (A.1)′–(A.3)′and (K.1)–(K.4)
hold. Moreover, assume that (Sn)Nattains its values in a compact setG˜ of R and fulfills Doeblin’s condition. If the
function R(x)=E(Yi|Xi=x) is continuous on G and if bn, hnsatisfy
bn=O
nhq n (logn)2M
n
, (3.1)
where Mnconverges arbitrary slowly to∞, then
| ˜Pn(1+)1−Pn(+1)1| →0 as n→ ∞, (3.2)
| ˜Pn(2+)1−Pn(+2)1| →0 as n→ ∞. (3.3)
Theorem 4. Assume that the conditions stated in Theorem 3 hold withδ> 1 in (A.2)′and (A.3)′. Then
| ˜Pn(3+)1−Pn(+3)1| →0 as n→ ∞. (3.4)
Moreover, if infx∈GE(Yi2|Xi =x) > supx∈GR(x) 2
,then
Proof of Theorem 3. Because of Theorem 3.4.7 in Györfi et al. (1989) the process (Sn)N is ϕ-mixing and its coefficients are of the form ϕn≤abn for some 0<b<1. According to Remark 3.3.3 in Györfi et al. (1989), the sequence (mn) obeys Condition (A.4) in Qian and Mammitzsch (1999). Applying Theorem 1 of Qian and Mammitzsch (1999) to the proof of Theorem 3.4.8 in Györfi et al. (1989) we get (3.2) and (3.3).
Proof of Theorem 4. Setδ′= 12(δ−1), thenδ′> 0. By (A.2)′and (A.3)′, we have E(|Yi2|1+δ′)=E(|Yi|1+δ)≤M,
E(|Yi2|1+δ′|Xi =x)=E(|Yi|1+δ|Xi =x)≤V for all x∈ ˆG. It follows from Theorem 3 that
| ˜Rn(Xn−q+1)−R(Xn−q+1)| →0 as n→ ∞, | ˜R∗n(Xn−q+1)−R∗(Xn−q+1)| →0 as n→ ∞.
Note that R is bounded on G. Similar to the proof of Theorem 2 we can show Theorem 4.
4. The estimator of credibility premium with percentile principle
We use the method proposed by Truong and Stone (1992) to estimate Pn+1in the case of the percentile principle.
For positive numbersδn, n≥1, that tend to zero as n→ ∞, and x∈Rqdefine In(x)=
n
i|
(Si−q+1, . . . , Si) T
−x
< δn, q≤i≤n−1 o
,
wherekdkmeans the Euclidean norm. Denote
ˆ
θn(u)=(1−ε)−percentile of {Si|i∈In(u)}. The estimator ofPn(5+)1is given by
ˆ
Pn(5+)1= ˆθn(Sn−q+1, . . . , Sn). Denote
θ (x)=minnp
P (Sn+1≤p|(Sn−q+1, . . . , Sn) T
=x)≥1−εo.
Let U be a nonempty neighborhood of the origin of Rqand let Xi, Yi be defined as in (2.3).
Condition 1. There is a positive constant M0such that |θ (x)−θ (x′)| ≤M0
x−x′
for x, x′∈U.
Condition 2. The distribution of X1 is absolutely continuous and its density f is bounded away from zero and
infinity on U; i.e., there is a positive constant M1such that
M1−1≤f (x)≤M1 for x ∈U.
Condition 3. For j≥2 the conditional distribution of X1=x has a density fj(·|x), and there is a positive constant
M2such that
Condition 4. The conditional distribution of Y1when X1=x is absolutely continuous and its density g(y|x) is
bounded away from zero and infinity over a neighborhood ofθ(x); i.e., there are positive constantsε0and M3
such that
M3−1≤g(y|x)≤M3 holds for all y ∈(θ (x)−ε0, θ (x)+ε0) and x∈U.
Theorem 5. Suppose thatδn∼n−r, r=1/(2+q) and that Conditions 1–4 hold. Let (Xn)N attain its values in a
compact subset C of U and let (Sn)Nsatisfy Doeblin’s condition. Then there is a positive constant c such that
lim n P (| ˆP
(5) n+1−P
(5)
n+1| ≥c(n−1logn)r)=0.
Proof. Because of Theorem 3.4.7 in Györfi et al. (1989), the process (Sn)N isϕ-mixing and its coefficients are of the form
ϕn≤abn for some 0< a <∞ and 0< b <1. Since
α(n)≤2(ϕn)1/2≤2a1/2bn/2=O(ρn),
whereρ =√bandα(n) is defined as in Truong and Stone (1992), Condition 5(iii) in Truong and Stone (1992) holds. Obviously,
| ˆPn(5+)1−Pn(5+)1| ≤sup x∈C| ˆ
θn(x)−θ (x)|,
so that we get
{| ˆPn(+5)1−Pn(+5)1| ≥c(n−1logn)r} ⊂
sup x∈C| ˆ
θn(x)−θ (x)| ≥c(n−1logn)r
.
Note that Theorem 3 in Truong and Stone (1992) holds also for the (1−ε) conditional percentile. It follows from Theorem 3 in Truong and Stone (1992) that
P{| ˆPn(5+)1−Pn(+5)1| ≥c(n−1logn)r} ≤P
sup x∈C| ˆ
θn(x)−θ (x)| ≥c(n−1logn)r
→0, n→ ∞.
This finishes the proof of Theorem 5.
Acknowledgements
I am grateful to V. Mammitzsch for his valuable comments and other assistance during my stay at the Department of Mathematics and Computer Science, University of Marburg, Germany.
References
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Collomb, G., 1984. Proprietes de convergence presque complete du predicteur a noyau. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 66, 441–460.
Doob, J., 1953. Stochastic Processes. Wiley, New York.
Gerber, H.U., 1979. An Introduction to Mathematical Risk Theory. University of Pennsylvania, Philadelphia, PA. Györfi, L., Härdle, W., Sarda, P., Vieu, P., 1989. Nonparametric Curve Estimation from Time Series. Springer, Berlin.
Qian, W., Mammitzsch, V., 1999. Strong uniform convergence for the estimator of the regression function underϕ-mixing conditions. Bericht Nr. 61. Reihe Mathematik, Fachbereich Mathematik und Informatik, Philipps-Universität, Marburg.
