45 research outputs found
Confidence regions for the multinomial parameter with small sample size
Consider the observation of n iid realizations of an experiment with d>1
possible outcomes, which corresponds to a single observation of a multinomial
distribution M(n,p) where p is an unknown discrete distribution on {1,...,d}.
In many applications, the construction of a confidence region for p when n is
small is crucial. This concrete challenging problem has a long history. It is
well known that the confidence regions built from asymptotic statistics do not
have good coverage when n is small. On the other hand, most available methods
providing non-asymptotic regions with controlled coverage are limited to the
binomial case d=2. In the present work, we propose a new method valid for any
d>1. This method provides confidence regions with controlled coverage and small
volume, and consists of the inversion of the "covering collection"' associated
with level-sets of the likelihood. The behavior when d/n tends to infinity
remains an interesting open problem beyond the scope of this work.Comment: Accepted for publication in Journal of the American Statistical
Association (JASA
Étude spectrale minutieuse de processus moins indécis que les autres
International audienceIn this paper we are looking for quantitative estimates for the convergene to equilibrium of non reversible Markov processes, especialy in short times. The models studied are simple enough to get an explicit expression of the L2 distance betweeen the semigroup and the invariant measure throught time and to compare it with the corresponding reversible cases
Spectrum of non-Hermitian heavy tailed random matrices
Let (X_{jk})_{j,k>=1} be i.i.d. complex random variables such that |X_{jk}|
is in the domain of attraction of an alpha-stable law, with 0< alpha <2. Our
main result is a heavy tailed counterpart of Girko's circular law. Namely,
under some additional smoothness assumptions on the law of X_{jk}, we prove
that there exists a deterministic sequence a_n ~ n^{1/alpha} and a probability
measure mu_alpha on C depending only on alpha such that with probability one,
the empirical distribution of the eigenvalues of the rescaled matrix a_n^{-1}
(X_{jk})_{1<=j,k<=n} converges weakly to mu_alpha as n tends to infinity. Our
approach combines Aldous & Steele's objective method with Girko's Hermitization
using logarithmic potentials. The underlying limiting object is defined on a
bipartized version of Aldous' Poisson Weighted Infinite Tree. Recursive
relations on the tree provide some properties of mu_alpha. In contrast with the
Hermitian case, we find that mu_alpha is not heavy tailed.Comment: Expanded version of a paper published in Communications in
Mathematical Physics 307, 513-560 (2011
The ensemble of random Markov matrices
The ensemble of random Markov matrices is introduced as a set of Markov or
stochastic matrices with the maximal Shannon entropy. The statistical
properties of the stationary distribution pi, the average entropy growth rate
and the second largest eigenvalue nu across the ensemble are studied. It is
shown and heuristically proven that the entropy growth-rate and second largest
eigenvalue of Markov matrices scale in average with dimension of matrices d as
h ~ log(O(d)) and nu ~ d^(-1/2), respectively, yielding the asymptotic relation
h tau_c ~ 1/2 between entropy h and correlation decay time tau_c = -1/log|nu| .
Additionally, the correlation between h and and tau_c is analysed and is
decreasing with increasing dimension d.Comment: 12 pages, 6 figur
On Poincare and logarithmic Sobolev inequalities for a class of singular Gibbs measures
This note, mostly expository, is devoted to Poincar{\'e} and log-Sobolev
inequalities for a class of Boltzmann-Gibbs measures with singular interaction.
Such measures allow to model one-dimensional particles with confinement and
singular pair interaction. The functional inequalities come from convexity. We
prove and characterize optimality in the case of quadratic confinement via a
factorization of the measure. This optimality phenomenon holds for all beta
Hermite ensembles including the Gaussian unitary ensemble, a famous exactly
solvable model of random matrix theory. We further explore exact solvability by
reviewing the relation to Dyson-Ornstein-Uhlenbeck diffusion dynamics admitting
the Hermite-Lassalle orthogonal polynomials as a complete set of
eigenfunctions. We also discuss the consequence of the log-Sobolev inequality
in terms of concentration of measure for Lipschitz functions such as maxima and
linear statistics.Comment: Minor improvements. To appear in Geometric Aspects of Functional
Analysis -- Israel Seminar (GAFA) 2017-2019", Lecture Notes in Mathematics
225
Circular Law Theorem for Random Markov Matrices
Consider an nxn random matrix X with i.i.d. nonnegative entries with bounded
density, mean m, and finite positive variance sigma^2. Let M be the nxn random
Markov matrix with i.i.d. rows obtained from X by dividing each row of X by its
sum. In particular, when X11 follows an exponential law, then M belongs to the
Dirichlet Markov Ensemble of random stochastic matrices. Our main result states
that with probability one, the counting probability measure of the complex
spectrum of n^(1/2)M converges weakly as n tends to infinity to the uniform law
on the centered disk of radius sigma/m. The bounded density assumption is
purely technical and comes from the way we control the operator norm of the
resolvent.Comment: technical update via http://HAL.archives-ouvertes.f
Dimension dependent hypercontractivity for Gaussian kernels
We derive sharp, local and dimension dependent hypercontractive bounds on the
Markov kernel of a large class of diffusion semigroups. Unlike the dimension
free ones, they capture refined properties of Markov kernels, such as trace
estimates. They imply classical bounds on the Ornstein-Uhlenbeck semigroup and
a dimensional and refined (transportation) Talagrand inequality when applied to
the Hamilton-Jacobi equation. Hypercontractive bounds on the Ornstein-Uhlenbeck
semigroup driven by a non-diffusive L\'evy semigroup are also investigated.
Curvature-dimension criteria are the main tool in the analysis.Comment: 24 page
Heisenberg Uncertainty Relation for Coarse-grained Observables
We ask which is the best strategy to reveal uncertainty relations between
comple- mentary observables of a continuous variable system for coarse-grained
measurements. This leads to the derivation of new uncertainty relations for
coarse-grained measurements that are always valid, even for detectors with low
precision. These relations should be particularly relevant in experimental
demonstrations of squeezing in quantum optics, quantum state reconstruction,
and the development of trustworthy entanglement criteria
Intertwining relations for one-dimensional diffusions and application to functional inequalities
International audienceFollowing the recent work [13] fulfilled in the discrete case, we pro- vide in this paper new intertwining relations for semigroups of one-dimensional diffusions. Various applications of these results are investigated, among them the famous variational formula of the spectral gap derived by Chen and Wang [15] together with a new criterion ensuring that the logarithmic Sobolev inequality holds. We complete this work by revisiting some classical examples, for which new estimates on the optimal constants are derived
A new method for the estimation of variance matrix with prescribed zeros in nonlinear mixed effects models
We propose a new method for the Maximum Likelihood Estimator (MLE) of
nonlinear mixed effects models when the variance matrix of Gaussian random
effects has a prescribed pattern of zeros (PPZ). The method consists in
coupling the recently developed Iterative Conditional Fitting (ICF) algorithm
with the Expectation Maximization (EM) algorithm. It provides positive definite
estimates for any sample size, and does not rely on any structural assumption
on the PPZ. It can be easily adapted to many versions of EM.Comment: Accepted for publication in Statistics and Computin