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
First we use a method proposed by Collomb 1984 to estimate ES
n +1
|S
n −q+1
, . . . , S
n
and ES
2 n
+1
| S
n −q+1
, . . . , S
n
. Denote
R
n
u =
P
n −1
i =q
S
i +1
Ku − S
i −q+1
, . . . , S
i T
h
n
P
n −1
i =q
Ku − S
i −q+1
, . . . , S
i T
h
n
, u
∈ E
q
, 2.1
R
∗ n
u =
P
n −1
i =q
S
2 i
+1
Ku − S
i −q+1
, . . . , S
i T
h
n
P
n −1
i =q
Ku − S
i −q+1
, . . . , S
i T
h
n
, u
∈ E
q
, 2.2
where K is a kernel function on R
q
with u ∈ E
q
and h
n
∈ R a bandwidth, h
n
0. R
n
S
n −q+1
, . . . , S
n
and R
∗ n
S
n −q+1
, . . . , S
n
are the estimators of ES
n +1
|S
n −q+1
, . . . , S
n
and ES
2 n
+1
|S
n −q+1
, . . . , S
n
, respectively.
W. Qian Insurance: Mathematics and Economics 27 2000 169–176 171
We establish the estimator of P
n +1
using the following principles. 1. Net premium principle:
ˆ P
n +1
= R
n
S
n −q+1
, . . . , S
n
. 2. Expected value principle:
ˆ P
n +1
= 1 + λR
n
S
n −q+1
, . . . , S
n
. 3. Variance principle:
ˆ P
n +1
= R
n
S
n −q+1
, . . . , S
n
+ α j
R
∗ n
S
n −q+1
, . . . , S
n
− R
n
S
n −q+1
, . . . , S
n 2
k .
4. Standard deviation principle: ˆ
P
n +1
= R
n
S
n −q+1
, . . . , S
n
+ βR
∗ n
S
n −q+1
, . . . , S
n
− R
n
S
n −q+1
, . . . , S
n 2
12
. 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 S
n N
is a stationary Markov process of order q, and denote by , A its state space. For x ∈
and for A ∈ A, denote by P
m
x, A the m-step transition probability function of S
n N
. The process S
n N
is said to satisfy Doeblin’s condition if there exists a finite measure ψ defined on A with ψ 0, a positive integer m and
some η 0 such that for each A ∈ A with ψA η there holds
P
m
x, A ≤ 1 − η for every x ∈ .
In short, we write X
i
≡ S
i
, . . . , S
i +q−1
T
, Y
i
≡ S
i +q
, 2.3
and introduce some assumptions on X
i
, Y
i
and the kernel function K. Let us denote a compact subset of R
q
by G and a compact ε-neighborhood of G ⊂ ˆ
G by ˆ G:
A.1 There are positive constants Ŵ, ν such that P X
i
∈ B ≤ ŴλB for every i ∈ N and all B ∈ BR
q
, P X
i
∈ B ≥ νλB for every i ∈ N and all B ∈ B ˆ G,
where BR
q
resp. B ˆ G is the σ -algebra of the Borel sets on R
q
resp. on ˆ G and λ the Lebesgue measure
on R
q
. A.2 There is a positive constant M such that
E |Y
i
|
β
≤ M for some β 2 and for every i ∈ N. A.3 There is a positive constant V such that
EY
i
− RX
i 2
|X
i
= x ≤ V for every i
∈ N and x ∈ ˆ G,
where RX
i
=EY
i
|X
i
. K.1 There is a positive constant ¯
K such that |Kx| ≤ ¯
K ∞ for every x ∈ R
q
. K.2
kxk
q
Kx → 0 as kxk → ∞,
where k
d
k denotes the Euclidean norm.
172 W. Qian Insurance: Mathematics and Economics 27 2000 169–176
K.3 There exists a positive constant ˆ K such that
Z
R
q
Ku du ≤
Z
R
q
|Ku| du ≤ ˆ K
∞. K.4 K is Lipschitz continuous of order γ on R
q
, 0 γ 1.
Theorem 1. Let S
n N
attain its values in a compact set ˜ G of R and satisfy Doeblin’s condition. Assume that
A.1–A.3 are satisfied by the variables X
i
, Y
i
defined as in 2.3 and that the kernel function satisfies K.1–K.4. Suppose that the bandwidth h
n
satisfies nh
q n
log
2
n → ∞,
n → ∞.
Then we have | ˆ
P
1 n
+1
− P
1 n
+1
| → 0 as n → ∞ and
| ˆ P
2 n
+1
− P
2 n
+1
| → 0 as n → ∞. To study the properties of ˆ
P
3 n
+1
and ˆ P
4 n
+1
we need a further condition. A.3
∗
There is a positive constant V
∗
such that EY
2 i
− R
∗
X
i 2
|X
i
= x ≤ V
∗
for every i ∈ N and x ∈ ˆ
G, where R
∗
X
i
= EY
2 i
|X
i
.
Theorem 2. Assume that the conditions stated in Theorem 1 hold. If A.3
∗
holds, A.2 holds for β 4 and sup
x ∈G
|Rx| ∞, where Rx = EY
i
|X
i
= x, then | ˆ
P
3 n
+1
− P
3 n
+1
| → 0 as n → ∞. 2.4
Moreover, if inf
x ∈G
EY
2 i
|X
i
= x sup
x ∈G
Rx
2
, then | ˆ
P
4 n
+1
− P
4 n
+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
| ˆ P
3 n
+1
− P
3 n
+1
| = |R
n
X
n −q+1
+ α[R
∗ n
X
n −q+1
− R
n
X
n −q+1
2
] − RX
n −q+1
− α[R
∗
X
n −q+1
−RX
n −q+1
2
] | ≤ |R
n
X
n −q+1
− RX
n −q+1
| + α|R
∗ n
X
n −q+1
− R
∗
X
n −q+1
| +α|R
n
X
n −q+1
2
− RX
n −q+1
2
| ≡ I
1
+ αI
2
+ αI
3
. 2.6
It follows from Theorem 3.4.8 in Györfi et al. 1989 that I
1
→ 0 as n → ∞, 2.7
I
2
→ 0 as n → ∞. 2.8
We still have to prove I
3
→ 0 as n → ∞.
W. Qian Insurance: Mathematics and Economics 27 2000 169–176 173
I
3
= I
1
|R
n
X
n −q+1
+ RX
n −q+1
| ≤ I
1
I
1
+ 2|RX
n −q+1
| ≤ I
1
I
1
+ 2 sup
x ∈G
|Rx| → 0 as n → ∞.
2.9 A combination of 2.6–2.9 gives 2.4.
Obviously, | ˆ
P
4 n
+1
− P
4 n
+1
| = |R
n
X
n −q+1
+ βR
∗ n
X
n −q+1
− R
n
X
n −q+1
2 12
− RX
n −q+1
− βR
∗
X
n −q+1
−RX
n −q+1
2 12
| ≤ |R
n
X
n −q+1
− RX
n −q+1
| +β
|R
∗ n
X
n −q+1
− R
∗
X
n −q+1
| + |R
n
X
n −q+1
2
− RX
n −q+1
2
| R
∗ n
X
n −q+1
− R
n
X
n −q+1
2 12
+ R
∗
X
n −q+1
− RX
n −q+1
2 12
≡ I
1
+ β I
2
+ I
3
R
∗ n
X
n −q+1
− R
n
X
n −q+1
2 12
+ R ∗ X
n −q+1
− RX
n −q+1
2 12
. 2.10
Since, by assumption, R
∗
X
n −q+1
− RX
n −q+1
2
≥ R
∗
X
n −q+1
− sup
x ∈G
Rx
2
≥ inf
x ∈G
EY
2 i
|X
i
= x − sup
x ∈G
Rx
2
0, 2.11
for large values of n we find R
∗ n
X
n −q+1
− R
n
X
n −q+1
2
≥ R
∗ n
X
n −q+1
− R
n
X
n −q+1
2
− R
∗
X
n −q+1
+ RX
n −q+1
2
+ inf
x ∈G
EY
2 i
|X
i
= x −
sup
x ∈G
Rx
2
≥ −I
2
− I
3
+ inf
x ∈G
EY
2 i
|X
i
= x − sup
x ∈G
Rx
2
≥ 1
2 inf
x ∈G
EY
2 i
|X
i
= x − sup
x ∈G
Rx
2
0. 2.12
Combining 2.10–2.12, we get | ˆ
P
4 n
+1
− P
4 n
+1
| ≤ I
1
+ β I
2
+ I
3
inf
x ∈G
EY
2 i
|X
i
= x − sup
x ∈G
Rx
2 12
→ 0 as n → ∞.
3. Improved kernel estimators of the credibility premium
