Page 197 - Contributed Paper Session (CPS) - Volume 6

CPS1881 Mike S.C. et al.
∗
∗
′′
∗
′′
for ∈ (0, ) , ( ) = 0 and () > 0 for ∈ ( , 1] . By mean-value
theroem, there are { , } such that
2
1
∗
′
( ) = 0 = ( ), 0 < < < < 1,
′
1
2
2
1
′
⁄
where () = −1 () . The maximum and minimum of () occur at

1
and , respectively, and
2
−1 () > −1 () , 0 < < ; < < 1
1 2
and
−1 () < −1 () , ≤ ≤ . (2.3)
1
2

The sample analog of { , } are solutions to the equations
2
1
̂
̂ −1
−1 (̂ ) − (̂ ) = 0, = 1, 2;

̅
̅

and the next two inequalities hold
̂
−1 () > ̂ −1 () , 0 < < ̂ ; ̂ < < 1
̅
̅
1 2
and
(2.4)
̂
̂ −1
−1 () < () , ̂ ≤ ≤ ̂ .
̅
̅
1 2
Case 2. The hypothesis was rarely discussed in the literature, because it
02
rarely occurs in practice. For ease of exposition, this Case 2 will not be
discussed because it would not make any significant effect of inference when
it would occur with negligibly small probability in application.
Under , it follows from the discussions above that
1

̂
−1 () > ̂ −1 () , 0 < < ̂
̅
̅
1
and (2.5)

̂
̂ −1
−1 () < () , ̂ < < 1.
̅
̅
1
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