20,991 research outputs found
Optimal rates of convergence for estimating the null density and proportion of nonnull effects in large-scale multiple testing
An important estimation problem that is closely related to large-scale
multiple testing is that of estimating the null density and the proportion of
nonnull effects. A few estimators have been introduced in the literature;
however, several important problems, including the evaluation of the minimax
rate of convergence and the construction of rate-optimal estimators, remain
open. In this paper, we consider optimal estimation of the null density and the
proportion of nonnull effects. Both minimax lower and upper bounds are derived.
The lower bound is established by a two-point testing argument, where at the
core is the novel construction of two least favorable marginal densities
and . The density is heavy tailed both in the spatial and frequency
domains and is a perturbation of such that the characteristic
functions associated with and match each other in low frequencies.
The minimax upper bound is obtained by constructing estimators which rely on
the empirical characteristic function and Fourier analysis. The estimator is
shown to be minimax rate optimal. Compared to existing methods in the
literature, the proposed procedure not only provides more precise estimates of
the null density and the proportion of the nonnull effects, but also yields
more accurate results when used inside some multiple testing procedures which
aim at controlling the False Discovery Rate (FDR). The procedure is easy to
implement and numerical results are given.Comment: Published in at http://dx.doi.org/10.1214/09-AOS696 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Estimating the Null and the Proportion of non-Null effects in Large-scale Multiple Comparisons
An important issue raised by Efron in the context of large-scale multiple
comparisons is that in many applications the usual assumption that the null
distribution is known is incorrect, and seemingly negligible differences in the
null may result in large differences in subsequent studies. This suggests that
a careful study of estimation of the null is indispensable.
In this paper, we consider the problem of estimating a null normal
distribution, and a closely related problem, estimation of the proportion of
non-null effects. We develop an approach based on the empirical characteristic
function and Fourier analysis. The estimators are shown to be uniformly
consistent over a wide class of parameters. Numerical performance of the
estimators is investigated using both simulated and real data. In particular,
we apply our procedure to the analysis of breast cancer and HIV microarray data
sets. The estimators perform favorably in comparison to existing methods.Comment: 42 pages, 6 figure
Holant Problems for Regular Graphs with Complex Edge Functions
We prove a complexity dichotomy theorem for Holant Problems on 3-regular
graphs with an arbitrary complex-valued edge function. Three new techniques are
introduced: (1) higher dimensional iterations in interpolation; (2) Eigenvalue
Shifted Pairs, which allow us to prove that a pair of combinatorial gadgets in
combination succeed in proving #P-hardness; and (3) algebraic symmetrization,
which significantly lowers the symbolic complexity of the proof for
computational complexity. With holographic reductions the classification
theorem also applies to problems beyond the basic model.Comment: 19 pages, 4 figures, added proofs for full versio
Finite -connected homogeneous graphs
A finite graph \G is said to be {\em -connected homogeneous}
if every isomorphism between any two isomorphic (connected) subgraphs of order
at most extends to an automorphism of the graph, where is a
group of automorphisms of the graph. In 1985, Cameron and Macpherson determined
all finite -homogeneous graphs. In this paper, we develop a method for
characterising -connected homogeneous graphs. It is shown that for a
finite -connected homogeneous graph \G=(V, E), either G_v^{\G(v)} is
--transitive or G_v^{\G(v)} is of rank and \G has girth , and
that the class of finite -connected homogeneous graphs is closed under
taking normal quotients. This leads us to study graphs where is
quasiprimitive on . We determine the possible quasiprimitive types for
in this case and give new constructions of examples for some possible types
Automorphisms of surfaces of general type with q>=2 acting trivially in cohomology
A compact complex manifold X is said to be rationally cohomologically
rigidified if its automorphism group Aut(X) acts faithfully on the cohomology
ring H*(X,Q). In this note, we prove that, surfaces of general type with
irregularity q>2 are rationally cohomologically rigidified, and so are minimal
surfaces S with q=2 unless K^2=8X. This answers a question of Fabrizio Catanese
in part.
As examples we give a complete classification of surfaces isogenous to a
product with q=2 that are not rationally cohomologically rigidified. These
surfaces turn out however to be rigidified.Comment: 18 pages; a remark and a closely relevant reference are adde
New Planar P-time Computable Six-Vertex Models and a Complete Complexity Classification
We discover new P-time computable six-vertex models on planar graphs beyond
Kasteleyn's algorithm for counting planar perfect matchings. We further prove
that there are no more: Together, they exhaust all P-time computable six-vertex
models on planar graphs, assuming #P is not P. This leads to the following
exact complexity classification: For every parameter setting in
for the six-vertex model, the partition function is either (1) computable in
P-time for every graph, or (2) #P-hard for general graphs but computable in
P-time for planar graphs, or (3) #P-hard even for planar graphs. The
classification has an explicit criterion. The new P-time cases in (2) provably
cannot be subsumed by Kasteleyn's algorithm. They are obtained by a non-local
connection to #CSP, defined in terms of a "loop space".
This is the first substantive advance toward a planar Holant classification
with not necessarily symmetric constraints. We introduce M\"obius
transformation on as a powerful new tool in hardness proofs for
counting problems.Comment: 61 pages, 16 figures. An extended abstract appears in SODA 202
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