Optimality

64 C. Cheng
Finally Pr (∩j∈J{Pj > α})≥ �
j∈J Pr(Pj > α), following from Jensen’s inequality.
The above considerations lead to the following definition.
Definition 4.1. The set of P values Pm has the positive orthant dependence property
if for any α∈[0, 1]
�
m�
�
m�
Pr {Pi > α} ≥ Pr (Pi > α).
i=1
This type of dependence is similar to the positive quadrant dependence introduced
by Lehmann [20].
Now define the upper envelope of the cdf’s of the P values as
i=1
F m(t) := max
i=1,...,m {Gi(t)}, t∈[0,1],
where Gi is the cdf of Pi. If Pm has the positive orthant dependence property then
�
m�
�
m�
Pr (P1:m≤ α)=1−Pr {Pi > α} ≤1− Pr (Pi > α)≤1−(1−F m(α)) m ,
implying
(4.4)
ERR(α ∗ cal)≤
i=1
i=1
π0α∗ cal
π0α∗ cal + (−π0)Hm(α ∗ cal )
�
1−(1−F m(α ∗ cal)) m� .
Because α∗ cal−→ 0 as m−→∞, the asymptotic magnitude of the above ERR can
be established by considering the magnitude of F m(tm) and Hm(tm) as tm−→ 0.
The following definition makes this idea rigorous.
Definition 4.2. The set of m P values Pm is said to be asymptotically stable as
m−→∞ if there exists sequences{βm},{ηm},{ψm},{ξm} and constants β ∗ , β∗,
η, ψ ∗ , ψ∗, and ξ such that
and
for sufficiently large m.
F m(t)�βmt ηm , Hm(t)�ψmt ξm , t−→ 0
0 < β∗≤ βm≤ β ∗