Optimality

330 R. M. Mnatsakanov and F. H. Ruymgaart
and
(5.8) E L 2 i = W
2 √ π
√ α
x S(x) + o(√ α), as α→∞,
respectively. So that combining (5.7) and (5.8) yields (5.6).
Corollary 5.1. If the parameter α = α(x) is chosen locally for each x > 0 as
follows
(5.9) α(x) = n 2/5 π
·{
4·W 2}1/5 · x 2 �
f
·
′ (x)
�
1−F(x)
then the estimator (5.4) with α = α(x) satisfies
MSE{ ˆ Sα(x)} = n −4/5
� 4/5
� �2/5
2 ′ 2
W · f (x)·(1−F(x))
π √ 2
, f ′ (x)�= 0.
+ o(1), as n→∞.
Theorem 5.2. Under the assumptions (2.3) and α = α(n)∼n δ for any 0 < δ < 2
we have, as n→∞,
(5.10)
ˆSα(x)−E ˆ Sα(x)
�
Var ˆ Sα(x)
→d Normal(0,1).
Theorem 5.3. Under the assumptions (2.3) we have
(5.11)
n1/2 α1/4{ ˆ �
Sα(x)−S(x)}→d Normal 0,
as n→∞, provided that we take α = α(n)∼n δ for any 2
5
W· S(x)
2 x √ �
,
π
< δ < 2.
Corollary 5.2. If the parameter α = α(x) is chosen locally for each x > 0 according
to (5.9) then for ˆ Sα(x) defined in (5.4) we have
n1/2 α(x) 1/4{ ˆ �
W S(x)
Sα(x)−S(x)}→d Normal −[
2 x √ π ]1/2 ,
provided f ′ (x)�= 0 and n→∞.
W S(x)
2 x √ �
,
π
Corollary 5.3. If the parameter α = α ∗ (x) is chosen locally for each x > 0
according to
(5.12) α ∗ (x) = n δ π
·{
4·W 2}1/5 · x 2 �
f
·
′ (x)
�
1−F(x)
then for ˆ Sα∗(x) defined in (5.4) we have
n1/2 α∗ (x) 1/4{ ˆ Sα∗(x)−S(x)}→d �
Normal 0,
provided f ′ (x)�= 0 and n→∞.
� 4/5
, 2
5
W S(x)
2 x √ �
,
π
< δ < 2 ,
Note that the proofs of all statements from Theorems 5.2 and 5.3 are similar to
the ones from Theorems 4.1 and 4.2, respectively.