104,128 research outputs found
The lower envelope of positive self-similar Markov processes
We establish integral tests and laws of the iterated logarithm for the lower
envelope of positive self-similar Markov processes at 0 and . Our
proofs are based on the Lamperti representation and time reversal arguments.
These results extend laws of the iterated logarithm for Bessel processes due to
Dvoretsky and Erd\"{o}s, Motoo and Rivero
Geometry of rank tests
We study partitions of the symmetric group which have desirable geometric
properties. The statistical tests defined by such partitions involve counting
all permutations in the equivalence classes. These permutations are the linear
extensions of partially ordered sets specified by the data. Our methods refine
rank tests of non-parametric statistics, such as the sign test and the runs
test, and are useful for the exploratory analysis of ordinal data. Convex rank
tests correspond to probabilistic conditional independence structures known as
semi-graphoids. Submodular rank tests are classified by the faces of the cone
of submodular functions, or by Minkowski summands of the permutohedron. We
enumerate all small instances of such rank tests. Graphical tests correspond to
both graphical models and to graph associahedra, and they have excellent
statistical and algorithmic properties.Comment: 8 pages, 4 figures. See also http://bio.math.berkeley.edu/ranktests/.
v2: Expanded proofs, revised after reviewer comment
On the future infimum of positive self-similar Markov processes
We establish integral tests and laws of the iterated logarithm for the upper
envelope of the future infimum of positive self-similar Markov processes and
for increasing self-similar Markov processes at 0 and infinity. Our proofs are
based on the Lamperti representation and time reversal arguments due to
Chaumont and Pardo [9]. These results extend laws of the iterated logarithm for
the future infimum of Bessel processes due to Khoshnevisan et al. [11]
Conjectures, tests and proofs: An overview of theory exploration
A key component of mathematical reasoning is the ability to formulate interesting conjectures about a problem domain at hand. In this paper, we give a brief overview of a theory exploration system called QuickSpec, which is able to automatically discover interesting conjectures about a given set of functions. QuickSpec works by interleaving term generation with random testing to form candidate conjectures. This is made tractable by starting from small sizes and ensuring that only terms that are irreducible with respect to already discovered conjectures are considered. QuickSpec has been successfully applied to generate lemmas for automated inductive theorem proving as well as to generate specifications of functional programs. We give an overview of typical use-cases of QuickSpec, as well as demonstrating how to easily connect it to a theorem prover of the user’s choice
Computationally Tractable Algorithms for Finding a Subset of Non-defective Items from a Large Population
In the classical non-adaptive group testing setup, pools of items are tested
together, and the main goal of a recovery algorithm is to identify the
"complete defective set" given the outcomes of different group tests. In
contrast, the main goal of a "non-defective subset recovery" algorithm is to
identify a "subset" of non-defective items given the test outcomes. In this
paper, we present a suite of computationally efficient and analytically
tractable non-defective subset recovery algorithms. By analyzing the
probability of error of the algorithms, we obtain bounds on the number of tests
required for non-defective subset recovery with arbitrarily small probability
of error. Our analysis accounts for the impact of both the additive noise
(false positives) and dilution noise (false negatives). By comparing with the
information theoretic lower bounds, we show that the upper bounds on the number
of tests are order-wise tight up to a factor, where is the number
of defective items. We also provide simulation results that compare the
relative performance of the different algorithms and provide further insights
into their practical utility. The proposed algorithms significantly outperform
the straightforward approaches of testing items one-by-one, and of first
identifying the defective set and then choosing the non-defective items from
the complement set, in terms of the number of measurements required to ensure a
given success rate.Comment: In this revision: Unified some proofs and reorganized the paper,
corrected a small mistake in one of the proofs, added more reference
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