3,426 research outputs found
The Principle of Open Induction on Cantor space and the Approximate-Fan Theorem
The paper is a contribution to intuitionistic reverse mathematics. We work in
a weak formal system for intuitionistic analysis. The Principle of Open
Induction on Cantor space is the statement that every open subset of Cantor
space that is progressive with respect to the lexicographical ordering of
Cantor space coincides with Cantor space. The Approximate-Fan Theorem is an
extension of the Fan Theorem that follows from Brouwer's principle of induction
on bars in Baire space and implies the Principle of Open Induction on Cantor
space. The Principle of Open Induction in Cantor space implies the Fan Theorem,
but, conversely the Fan Theorem does not prove the Principle of Open Induction
on Cantor space. We list a number of equivalents of the Principle of Open
Induction on Cantor space and also a number of equivalents of the
Approximate-Fan Theorem
Brouwer's Fan Theorem as an axiom and as a contrast to Kleene's Alternative
The paper is a contribution to intuitionistic reverse mathematics. We
introduce a formal system called Basic Intuitionistic Mathematics BIM, and then
search for statements that are, over BIM, equivalent to Brouwer's Fan Theorem
or to its positive denial, Kleene's Alternative to the Fan Theorem. The Fan
Theorem is true under the intended intuitionistic interpretation and Kleene's
Alternative is true in the model of BIM consisting of the Turing-computable
functions. The task of finding equivalents of Kleene's Alternative is,
intuitionistically, a nontrivial extension of finding equivalents of the Fan
Theorem, although there is a certain symmetry in the arguments that we shall
try to make transparent.
We introduce closed-and-separable subsets of Baire space and of the set of
the real numbers. Such sets may be compact and also positively noncompact. The
Fan Theorem is the statement that Cantor space, or, equivalently, the unit
interval, is compact, and Kleene's Alternative is the statement that Cantor
space, or, equivalently, the unit interval is positively noncompact. The class
of the compact closed-and-separable sets and also the class of the
closed-and-separable sets that are positively noncompact are characterized in
many different ways and a host of equivalents of both the Fan Theorem and
Kleene's Alternative is found
On Two Simple and Effective Procedures for High Dimensional Classification of General Populations
In this paper, we generalize two criteria, the determinant-based and
trace-based criteria proposed by Saranadasa (1993), to general populations for
high dimensional classification. These two criteria compare some distances
between a new observation and several different known groups. The
determinant-based criterion performs well for correlated variables by
integrating the covariance structure and is competitive to many other existing
rules. The criterion however requires the measurement dimension be smaller than
the sample size. The trace-based criterion in contrast, is an independence rule
and effective in the "large dimension-small sample size" scenario. An appealing
property of these two criteria is that their implementation is straightforward
and there is no need for preliminary variable selection or use of turning
parameters. Their asymptotic misclassification probabilities are derived using
the theory of large dimensional random matrices. Their competitive performances
are illustrated by intensive Monte Carlo experiments and a real data analysis.Comment: 5 figures; 22 pages. To appear in "Statistical Papers
The Fan Theorem, its strong negation, and the determinacy of games
IIn the context of a weak formal theory called Basic Intuitionistic
Mathematics , we study Brouwer's Fan Theorem and a strong
negation of the Fan Theorem, Kleene's Alternative (to the Fan Theorem). We
prove that the Fan Theorem is equivalent to contrapositions of a number of
intuitionistically accepted axioms of countable choice and that Kleene's
Alternative is equivalent to strong negations of these statements. We also
discuss finite and infinite games and introduce a constructively useful notion
of determinacy. We prove that the Fan Theorem is equivalent to the
Intuitionistic Determinacy Theorem, saying that every subset of Cantor space
is, in our constructively meaningful sense, determinate, and show that Kleene's
Alternative is equivalent to a strong negation of a special case of this
theorem. We then consider a uniform intermediate value theorem and a
compactness theorem for classical propositional logic, and prove that the Fan
Theorem is equivalent to each of these theorems and that Kleene's Alternative
is equivalent to strong negations of them. We end with a note on a possibly
important statement, provable from principles accepted by Brouwer, that one
might call a Strong Fan Theorem.Comment: arXiv admin note: text overlap with arXiv:1106.273
Distributed linear regression by averaging
Distributed statistical learning problems arise commonly when dealing with
large datasets. In this setup, datasets are partitioned over machines, which
compute locally, and communicate short messages. Communication is often the
bottleneck. In this paper, we study one-step and iterative weighted parameter
averaging in statistical linear models under data parallelism. We do linear
regression on each machine, send the results to a central server, and take a
weighted average of the parameters. Optionally, we iterate, sending back the
weighted average and doing local ridge regressions centered at it. How does
this work compared to doing linear regression on the full data? Here we study
the performance loss in estimation, test error, and confidence interval length
in high dimensions, where the number of parameters is comparable to the
training data size. We find the performance loss in one-step weighted
averaging, and also give results for iterative averaging. We also find that
different problems are affected differently by the distributed framework.
Estimation error and confidence interval length increase a lot, while
prediction error increases much less. We rely on recent results from random
matrix theory, where we develop a new calculus of deterministic equivalents as
a tool of broader interest.Comment: V2 adds a new section on iterative averaging methods, adds
applications of the calculus of deterministic equivalents, and reorganizes
the pape
Generalized Disappointment Aversion and Asset Prices
We provide an axiomatic model of preferences over atemporal risks that generalizes Gul (1991) A Theory of Disappointment Aversion' by allowing risk aversion to be first order' at locations in the state space that do not correspond to certainty. Since the lotteries being valued by an agent in an asset-pricing context are not typically local to certainty, our generalization, when embedded in a dynamic recursive utility model, has important quantitative implications for financial markets. We show that the state-price process, or asset-pricing kernel, in a Lucas-tree economy in which the representative agent has generalized disappointment aversion preferences is consistent with the pricing kernel that resolves the equity-premium puzzle. We also demonstrate that a small amount of conditional heteroskedasticity in the endowment-growth process is necessary to generate these favorable results. In addition, we show that risk aversion in our model can be both state-dependent and counter-cyclical, which empirical research has demonstrated is necessary for explaining observed asset-pricing behavior.
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