Insurance: Mathematics and Economics 27 2000 169–176
An application of nonparametric regression estimation in credibility theory
q
Weimin Qian
a,b, ∗
a
Department of Mathematics and Computer Science, University of Marburg, 35032 Marburg, Germany
b
Department 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 principleprinciple of equivalence. This means that P = ES.
2. The expected value principle. This means that there is a safety loading proportional to ES. Thus P = 1 + λES,
where λ 0 is a parameter. 3. The variance principle. A safety loading proportional to the variance, i.e. P
= ES + α VarS, where α 0 is a parameter.
4. The standard deviation principle. A safety loading proportional to the standard deviation, i.e. P = ES +
β √
VarS, where β 0 is a parameter. 5. The percentile principlechance constrained premium. Let 0 ε 1. Then P is determined such that the prob-
ability for a loss from the contract is at most ε, i.e. P
= min{p|F p ≥ 1 − ε}, where F p
≡ P {S ≤ p}. Let S
1
, S
2
, . . . denote the claims from subsequent periods. Given S
1
, S
2
, . . . , S
n
, one has to determine an appropriate premium to cover the risk S
n +1
. Let us denote P
i
as the credibility premium to cover S
i
.
q
Research supported by an Allianz scholarship of Alexander von Humboldt Foundation.
∗
Tel.: +49-6421-2825484.
E-mail address: qianweiminhotmail.com W. Qian. 0167-668700 – see front matter © 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 7 - 6 6 8 7 0 0 0 0 0 4 4 - 5
170 W. Qian Insurance: Mathematics and Economics 27 2000 169–176
We suppose that S
1
, S
2
, . . . 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 P
n +1
= H S
n +1
|S
1
, S
2
, . . . , S
n
. We find the following equations by using various principles.
1. Net premium principle: P
1 n
+1
= ES
n +1
|S
1
, . . . , S
n
= ES
n +1
|S
n −q+1
, . . . , S
n
. 2. Expected value principle:
P
2 n
+1
= 1 + λES
n +1
|S
n −q+1
, . . . , S
n
. 3. Variance principle:
P
3 n
+1
= ES
n +1
|S
n −q+1
, . . . , S
n
+ α VarS
n +1
|S
n −q+1
, . . . , S
n
= ES
n +1
|S
n −q+1
, . . . , S
n
+ αES
2 n
+1
|S
n −q+1
, . . . , S
n
−ES
n +1
|S
n −q+1
, . . . , S
n 2
. 4. Standard deviation principle:
P
4 n
+1
= ES
n +1
|S
n −q+1
, . . . , S
n
+ βES
2 n
+1
|S
n −q+1
, . . . , S
n
− ES
n +1
|S
n −q+1
, . . . , S
n 2
12
. 5. Percentile principle:
P
5 n
+1
= min{p|F
n +1
p ≥ 1 − ε}.
where F
n +1
x ≡ P {S
n +1
≤ x|S
n −q+1
, . . . , S
n
}. In this paper we use the nonparametric regression method to establish estimators for P
i n
+1
, i = 1, . . . , 5. Kernel
estimators for P
i n
+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 for P
5 n
+1
introduced by Truong and Stone and examine its asymptotic property.
2. Kernel estimators of the credibility premium
