Optimality

328 R. M. Mnatsakanov and F. H. Ruymgaart
Corollary 4.1. Let us assume that (2.2) is valid. Consider � f ∗ α(x) defined in (3.15)
with α(x) given by (3.14). Then
(4.4)
n1/2 α(x) 1/4{ � f ∗ �
α(x)−f(x)}→d Normal [ W f(x)
as n→∞ and f ′′ (x)�= 0.
2 x2√π ]1/2 ,
W f(x)
2 x2√ �
,
π
Proof. From (4.1) and (3.11) with α = α(x) defined in (3.13) it is easy to see that
(4.5)
n1/2 α(x) 1/4{ � f ∗ α(x)−E � f ∗ �
α(x)} = Normal 0,
W f(x)
2 x2√ �
π
+ oP( 1
),
n2/5 as n→∞. Application of (3.4) where α = α(x) is defined by (3.14) yields (4.4).
Corollary 4.2. Let us assume that (2.2) is valid. Consider � f ∗ α∗(x) defined in (3.15)
with α∗ (x) given by
(4.6) α ∗ (x) = n δ ·{
π
4·W 2}1/5
�
x3 · f ′′ �4/5 (x)
� ,
f(x)
2
5
Then when f ′′ (x)�= 0, and letting n→∞, it follows that
(4.7)
n1/2 α∗ (x) 1/4{ � f ∗ α∗(x)−f(x)}→d �
Normal 0,
n1/2 α∗ (x) 1/4{ � f ∗ α∗(x)−E � f ∗ �
α∗(x)} = Normal 0,
< δ < 2.
W f(x)
2 x2√ �
.
π
Proof. Again from (4.1) and (3.11) with α = α∗ (x) defined in (4.6) it is easy to see
that
W f(x)
(4.8)
2 x2√ �
+ oP(1),
π
as n→∞. On the other hand application of (3.4) where α = α ∗ (x) is defined by
(4.6) yields
(4.9)
n1/2 α∗ (x) 1/4{E � f ∗ �
C(x)
α∗(x)−f(x)} = O
n (5δ−2)/4
�
,
W f(x)
as n→∞. Here C(x) ={ 2 x2 √ π }1/2 . Combining (4.8) and (4.9) yields (4.7).
5. An application to the excess life distribution
Assume that the random variable X has cdf F and pdf f defined on [0,∞) with
F(0) = 0. Denote the hazard rate function hF = f/S, where S = 1−F is the
corresponding survival function of X. Assume also that the sampled density g
satisties (1.1) and (1.7). It follows that
(5.1) g(y) = 1
W
{1−F(y)} , y≥ 0 .
It is also immediate that W = 1/g(0) and, f(y) =−W g ′ (y) =− g′ (y)
g(0)
that
hF(y) =− g′ (y)
g(y)
, y≥ 0 .
, y≥0 , so