3,774 research outputs found

### Universal Coding on Infinite Alphabets: Exponentially Decreasing Envelopes

This paper deals with the problem of universal lossless coding on a countable
infinite alphabet. It focuses on some classes of sources defined by an envelope
condition on the marginal distribution, namely exponentially decreasing
envelope classes with exponent $\alpha$. The minimax redundancy of
exponentially decreasing envelope classes is proved to be equivalent to
$\frac{1}{4 \alpha \log e} \log^2 n$. Then a coding strategy is proposed, with
a Bayes redundancy equivalent to the maximin redundancy. At last, an adaptive
algorithm is provided, whose redundancy is equivalent to the minimax redundanc

### Bernstein von Mises Theorems for Gaussian Regression with increasing number of regressors

This paper brings a contribution to the Bayesian theory of nonparametric and
semiparametric estimation. We are interested in the asymptotic normality of the
posterior distribution in Gaussian linear regression models when the number of
regressors increases with the sample size. Two kinds of Bernstein-von Mises
Theorems are obtained in this framework: nonparametric theorems for the
parameter itself, and semiparametric theorems for functionals of the parameter.
We apply them to the Gaussian sequence model and to the regression of functions
in Sobolev and $C^{\alpha}$ classes, in which we get the minimax convergence
rates. Adaptivity is reached for the Bayesian estimators of functionals in our
applications

### Smooth times of a flow in dimension 1

Let $\alpha$ be an irrational number and $I$ an interval of $\mathbb{R}$. If
$\alpha$ is diophantine, we show that any one-parameter group of homeomorphisms
of $I$ whose time-$1$ and $\alpha$ maps are $C^\infty$ is in fact the flow of a
$C^\infty$ vector field. If $\alpha$ is Liouville on the other hand, we
construct a one-parameter group of homeomorphisms of $I$ whose time-$1$ and
$\alpha$ maps are $C^\infty$ but which is not the flow of a $C^2$ vector field
(though, if $I$ has boundary, we explain that the hypotheses force it to be the
flow of a $C^1$ vector field). We extend both results to families of irrational
numbers, the critical arithmetic condition in this case being simultaneous
"diophantinity". For one-parameter groups defining a free action of
$(\mathbb{R},+)$ on $I$, these results follow from famous linearization
theorems for circle diffeomorphisms. The novelty of this work concerns non-free
actions.Comment: 35 pages, 8 figure

### On the connectedness of the space of codimension one foliations on a closed 3-manifold

We study the topology of the space of smooth codimension one foliations on a
closed 3-manifold. We regard this space as the space of integrable plane fields
included in the space of all smooth plane fields. It has been known since the
late 60's that every plane field can be deformed continuously to an integrable
one, so the above inclusion induces a surjective map between connected
components. We prove that this map is actually a bijection.Comment: 47 pages, 22 figure

### Clustering and variable selection for categorical multivariate data

This article investigates unsupervised classification techniques for
categorical multivariate data. The study employs multivariate multinomial
mixture modeling, which is a type of model particularly applicable to
multilocus genotypic data. A model selection procedure is used to
simultaneously select the number of components and the relevant variables. A
non-asymptotic oracle inequality is obtained, leading to the proposal of a new
penalized maximum likelihood criterion. The selected model proves to be
asymptotically consistent under weak assumptions on the true probability
underlying the observations. The main theoretical result obtained in this study
suggests a penalty function defined to within a multiplicative parameter. In
practice, the data-driven calibration of the penalty function is made possible
by slope heuristics. Based on simulated data, this procedure is found to
improve the performance of the selection procedure with respect to classical
criteria such as BIC and AIC. The new criterion provides an answer to the
question "Which criterion for which sample size?" Examples of real dataset
applications are also provided

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