26,187 research outputs found
Perturbed Datasets Methods for Hypothesis Testing and Structure of Corresponding Confidence Sets
Hypothesis testing methods that do not rely on exact distribution assumptions
have been emerging lately. The method of sign-perturbed sums (SPS) is capable
of characterizing confidence regions with exact confidence levels for linear
regression and linear dynamical systems parameter estimation problems if the
noise distribution is symmetric. This paper describes a general family of
hypothesis testing methods that have an exact user chosen confidence level
based on finite sample count and without relying on an assumed noise
distribution. It is shown that the SPS method belongs to this family and we
provide another hypothesis test for the case where the symmetry assumption is
replaced with exchangeability. In the case of linear regression problems it is
shown that the confidence regions are connected, bounded and possibly
non-convex sets in both cases. To highlight the importance of understanding the
structure of confidence regions corresponding to such hypothesis tests it is
shown that confidence sets for linear dynamical systems parameter estimates
generated using the SPS method can have non-connected parts, which have far
reaching consequences
On the Complexity of Nondeterministically Testable Hypergraph Parameters
The paper proves the equivalence of the notions of nondeterministic and
deterministic parameter testing for uniform dense hypergraphs of arbitrary
order. It generalizes the result previously known only for the case of simple
graphs. By a similar method we establish also the equivalence between
nondeterministic and deterministic hypergraph property testing, answering the
open problem in the area. We introduce a new notion of a cut norm for
hypergraphs of higher order, and employ regularity techniques combined with the
ultralimit method.Comment: 33 page
Distributed Testing of Excluded Subgraphs
We study property testing in the context of distributed computing, under the
classical CONGEST model. It is known that testing whether a graph is
triangle-free can be done in a constant number of rounds, where the constant
depends on how far the input graph is from being triangle-free. We show that,
for every connected 4-node graph H, testing whether a graph is H-free can be
done in a constant number of rounds too. The constant also depends on how far
the input graph is from being H-free, and the dependence is identical to the
one in the case of testing triangles. Hence, in particular, testing whether a
graph is K_4-free, and testing whether a graph is C_4-free can be done in a
constant number of rounds (where K_k denotes the k-node clique, and C_k denotes
the k-node cycle). On the other hand, we show that testing K_k-freeness and
C_k-freeness for k>4 appear to be much harder. Specifically, we investigate two
natural types of generic algorithms for testing H-freeness, called DFS tester
and BFS tester. The latter captures the previously known algorithm to test the
presence of triangles, while the former captures our generic algorithm to test
the presence of a 4-node graph pattern H. We prove that both DFS and BFS
testers fail to test K_k-freeness and C_k-freeness in a constant number of
rounds for k>4
Faster Family-wise Error Control for Neuroimaging with a Parametric Bootstrap
In neuroimaging, hundreds to hundreds of thousands of tests are performed
across a set of brain regions or all locations in an image. Recent studies have
shown that the most common family-wise error (FWE) controlling procedures in
imaging, which rely on classical mathematical inequalities or Gaussian random
field theory, yield FWE rates that are far from the nominal level. Depending on
the approach used, the FWER can be exceedingly small or grossly inflated. Given
the widespread use of neuroimaging as a tool for understanding neurological and
psychiatric disorders, it is imperative that reliable multiple testing
procedures are available. To our knowledge, only permutation joint testing
procedures have been shown to reliably control the FWER at the nominal level.
However, these procedures are computationally intensive due to the increasingly
available large sample sizes and dimensionality of the images, and analyses can
take days to complete. Here, we develop a parametric bootstrap joint testing
procedure. The parametric bootstrap procedure works directly with the test
statistics, which leads to much faster estimation of adjusted \emph{p}-values
than resampling-based procedures while reliably controlling the FWER in sample
sizes available in many neuroimaging studies. We demonstrate that the procedure
controls the FWER in finite samples using simulations, and present region- and
voxel-wise analyses to test for sex differences in developmental trajectories
of cerebral blood flow
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