Optimality

326 R. M. Mnatsakanov and F. H. Ruymgaart
This yields for the bias (µ = x, σ 2 = x 2 /α)
(3.8)
� ∞
fα(x)−f(x) = hα,x,1(y){f(y)−f(x)}du
0
� ∞
= hα,x,1(y){f(x) + (y− x)f
0
′ (x)
+ 1
� ∞
(y− x)
2 0
2 {f ′′ (˜y)−f(x)}dy
= 1
� ∞
(y− x)
2 0
2 hα,x,1(y)f ′′ (x)du
+ 1
� ∞
2 0
= 1 x
2
2
α f ′′ � �
1
(x) + o , as α→∞.
α
For the variance we have
(y− x) 2 hα,x,1(y){f ′′ (˜y)−f ′′ (x)}dy
(3.9) Var � f ∗ α(x) = 1
n VarMi = 1
n {E M2 i− f 2 α(x)}.
Applying (3.6) for k = 2 yields
(3.10)
E M 2 i = W α2
x 2
∼ α2
x2 (2α−3)!
{(α−1)!} 2
1
22α−2 � ∞
hα,x,2(u)f(u)du
0
e−(2α−3) {(2α−3)} (2α−3)+1/2
e−2(α−1) {(α−1)} 2(α−1)+1
1
22(α−1) W
√
2π
� ∞
× hα,x,2(u)f(u)du =
0
W
2 √ √
α
π x2 � ∞
hα,x,2(u)f(u)du
0
= W
2 √ √
α
W
π x2{f(x) + o(1)} =
2 √ √
α
π x2 f(x) + o(√α) as α→∞. Now inserting this in (3.9) we obtain
(3.11)
Var � f ∗ α(x) = 1
�
W
n 2 √ π
= W√ α
2n √ π
Finally, this leads to the MSE of � f ∗ α(x):
(3.12) MSE{ � f ∗ α(x)} = W√ α
2n √ π
For optimal rate we may take
√
α
x2 f(x) + o(√α)− �√ �
f(x) α
+ o .
x2 n
�
f(x) + O
f(x) 1 x
+
x2 4
4
α2{f ′′ (x)} 2 + o
(3.13) α = αn∼ n 2/5 ,
�√ �
α
n
� ��2 1
�
α
�
1
+ o
α2 �
.
assuming that n is such that αn is an integer. By substitution (3.13) in (3.12) we
find (3.5).