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
In this section we study improved kernel estimators of P
i n
+1
, i = 1, . . . , 4.
Let b
n
, n ∈ N, be a sequence of positive numbers and h
n
, n ∈ N, a sequence of bandwidths. Consider the estimators
˜ R
n
u =
P
n −1
i =q
S
i +1
I
{|S
i +1
|b
n
}
Ku − X
i −q+1
h
n
P
n −1
i =q
Ku − X
i −q+1
h
n
, ˜
R
∗ n
= P
n −1
i =q
S
2 i
+1
I
{|S
i +1
| √
b
n
}
Ku − X
i −q+1
h
n
P
n −1
i =q
Ku − X
i −q+1
h
n
.
174 W. Qian Insurance: Mathematics and Economics 27 2000 169–176
We establish the improved kernel estimators of P
n +1
in the case of the following principles. 1. Net premium principle:
˜ P
1 n
+1
= ˜ R
n
X
n −q+1
. 2. Expected value principle:
˜ P
2 n
+1
= 1 + λ ˜ R
n
X
n −q+1
. 3. Variance principle:
˜ P
3 n
+1
= ˜ R
n
X
n −q+1
+ α ˜ R
∗ n
X
n −q+1
− ˜ R
n
X
n −q+1
2
. 4. Standard deviation principle:
˜ P
4 n
+1
= ˜ R
n
X
n −q+1
+ β ˜ R
∗ n
X
n −q+1
− ˜ R
n
X
n −q+1
2 12
. Here we use the notations of Section 2. Moreover, we state some new assumptions.
A.1
′
Assume the existence of a common marginal density f of the random variables x
i
. f is continuous on ˆ
G, and there is a positive constant m
1
such that f
x ≥ m
1
for every x ∈ G.
A.2
′
There exists a positive constant M such that E
|Y
i
|
1 +δ
≤ M for some δ 0 and all i ∈ N. A.3
′
There exists a positive constant V such that E
|Y
i
|
1 +δ
|X
i
= x ≤ V for some δ 0 and all x
∈ ˆ G, i
∈ N.
Theorem 3. Suppose that the variables X
i
, Y
i
defined as in 2.3 obey Conditions A.1
′
–A.3
′
and K.1–K.4 hold. Moreover, assume that S
n N
attains its values in a compact set ˜ G of R and fulfills Doeblin’s condition. If the
function Rx = EY
i
|X
i
= x is continuous on G and if b
n
, h
n
satisfy b
n
= O nh
q n
log n
2
M
n
, 3.1
where M
n
converges arbitrary slowly to ∞, then
| ˜ P
1 n
+1
− P
1 n
+1
| → 0 as n → ∞, 3.2
| ˜ P
2 n
+1
− P
2 n
+1
| → 0 as n → ∞. 3.3
Theorem 4. Assume that the conditions stated in Theorem 3 hold with δ 1 in A.2
′
and A.3
′
. Then | ˜
P
3 n
+1
− P
3 n
+1
| → 0 as n → ∞. 3.4
Moreover, if inf
x ∈G
EY
2 i
|X
i
= x sup
x ∈G
Rx
2
, then | ˜
P
4 n
+1
− P
4 n
+1
| → 0 as n → ∞. 3.5
W. Qian Insurance: Mathematics and Economics 27 2000 169–176 175
Proof of Theorem 3. Because of Theorem 3.4.7 in Györfi et al. 1989 the process S
n N
is ϕ-mixing and its coefficients are of the form ϕ
n
≤ ab
n
for some 0 b 1. According to Remark 3.3.3 in Györfi et al. 1989, the sequence m
n
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. Set δ
′
=
1 2
δ − 1, then δ
′
0. By A.2
′
and A.3
′
, we have E
|Y
2 i
|
1 +δ
′
= E|Y
i
|
1 +δ
≤ M, E
|Y
2 i
|
1 +δ
′
|X
i
= x = E|Y
i
|
1 +δ
|X
i
= x ≤ V for all x
∈ ˆ G.
It follows from Theorem 3 that | ˜
R
n
X
n −q+1
− RX
n −q+1
| → 0 as n → ∞, | ˜
R
∗ n
X
n −q+1
− R
∗
X
n −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
