Optimality

P value EDF
qdf
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.4 0.8 1.2 1.6 2.0
| ||| | | |
interior knot positions
Massive multiple hypotheses testing 61
(a)
0.0 0.2 0.4 0.6 0.8 1.0
(c)
0.0 0.2 0.4 0.6 0.8 1.0
u
P value EQF
EQF, Q^, and conv. backbone
0.0 0.2 0.4 0.6 0.8 1.0
(b)
||| | | | | | | | |
0.0 0.2 0.4 0.6 0.8 1.0
t* positions
0.0 0.2 0.4 0.6 0.8 1.0
Fig 1. (a) The interior knot positions indicated by | and the P value EDF; (b) the positions of t ∗ j
indicated by | and the P value EQF; (c) �qm: the derivative of �Qm; (d) the P value EQF (solid),
the smoothed EQF �Qm from Algorithm 1 (dash-dot), and the convex backbone �Q ∗ m (long dash).
a measurement that is an L 2 distance. The concept of convex backbone facilitates
the derivation of a measurement more adaptive to the ensemble distribution of the
P values. Given the convex backbone Q ∗ m(·) := Q ∗ m(·;γ, a, d, b1, b0, τm) as defined
in (3.2), the “model-fitting” term can be defined as the L γ distance between Q ∗ m(·)
and uniformity on [0, α]:
Dγ(α) :=
�� α
0
0.0 0.2 0.4 0.6 0.8 1.0
(t−Q ∗ m(t)) γ �1/γ dt , α∈(0,1].
The adaptivity is reflected by the use of the L γ distance: Recall that the larger
the γ, the higher concentration of small P values, and the norm inequality (Hardy,
Littlewood, and Pólya [16], P.157) implies that Dγ2(α)≥Dγ1(α) for every α∈(0,1]
if γ2 > γ1.
Clearly Dγ(α) is non-decreasing in α. Intuitively one possibility would be to
maximize a criterion like Dγ(α)−λπ0mα. However, the two terms are not on the
(d)
u