arXiv subjects

Silvia R. C. Lopes

Publications and source records attributed to Silvia R. C. Lopes.

5 recordsLinked to original sources

arXiv2024

Parametrized Families of Gibbs Measures and their Statistical Inference

For Hölder continuous functions $f_i$, $i=0,\ldots ,d$, on a subshift of finite type and $Θ\subset \mathbb \R^d$ we consider a parametrized family of potentials $\{F_θ= f_0+\sum_{i=1}^d θ_i f_i : θ\in Θ\}$. We show that the maximum likelihood estimator of $θ$ for a family of Gibbs measures with potentials $F_θ$ is consistent and determine its asymptotic distribution under the associated shift-invariant distribution. A second part discusses applications; from confidence intervals through testing problems to connections to Bernoulli distributions and stationary Markov chains.

math.DS↗

arXiv2022

Bayes posterior convergence for loss functions via almost additive Thermodynamic Formalism

Statistical inference can be seen as information processing involving input information and output information that updates belief about some unknown parameters. We consider the Bayesian framework for making inferences about dynamical systems from ergodic observations, where the Bayesian procedure is based on the Gibbs posterior inference, a decision process generalization of standard Bayesian inference where the likelihood is replaced by the exponential of a loss function. In the case of direct observation and almost-additive loss functions, we prove an exponential convergence of the a posteriori measures a limit measure. Our estimates on the Bayes posterior convergence for direct observation are related and extend those in a recent paper by K. McGoff, S. Mukherjee and A. Nobel. Our approach makes use of non-additive thermodynamic formalism and large deviation properties instead of joinings.

math.ST↗

arXiv2021

Decision Theory and Large Deviations for Dynamical Hypotheses Tests: Neyman-Pearson, Min-Max and Bayesian Tests

We analyze hypotheses tests using classical results on large deviations to compare two models, each one described by a different Hölder Gibbs probability measure. One main difference to the classical hypothesis tests in Decision Theory is that here the two measures are singular with respect to each other. Among other objectives, we are interested in the decay rate of the wrong decisions probability, when the sample size $n$ goes to infinity. We show a dynamical version of the Neyman-Pearson Lemma displaying the ideal test within a certain class of similar tests. This test becomes exponentially better, compared to other alternative tests, when the sample size goes to infinity. We are able to present the explicit exponential decay rate. We also consider both, the Min-Max and a certain type of Bayesian hypotheses tests. We shall consider these tests in the log likelihood framework by using several tools of Thermodynamic Formalism. Versions of the Stein's Lemma and Chernoff's information are also presented.

math.ST↗

arXiv2021

Pentadiagonal Matrices and an Application to the Centered MA(1) Stationary Gaussian Process

In this work, we study the properties of a pentadiagonal symmetric matrix with perturbed corners. More specifically, we present explicit expressions for characterizing when this matrix is non-negative and positive definite in two special and important cases. We also give a closed expression for the determinant of such matrices. Previous works present the determinant in a recurrence form but not in an explicit one. As an application of these results, we also study the limiting cumulant generating function associated to the bivariate sequence of random vectors (n^{-1} (\sum_{k=1}^n X_k^2 , \sum_{k=2}^n X_k X_{k-1})_{n in N}, when (X_n)_{n in N} is the centered stationary moving average process of first order with Gaussian innovations. We exhibit the explicit expression of this limiting cumulant generating function. Finally, we present three examples illustrating the techniques studied here.

math.PR↗

arXiv2007

Parameter Estimation in Manneville-Pomeau Processes

In this work we study a class of stochastic processes $\{X_t\}_{t\in\N}$, where $X_t = (ϕ\circ T_s^t)(X_0)$ is obtained from the iterations of the transformation T_s, invariant for an ergodic probability μ_s on [0,1] and a continuous by part function $ϕ:[0,1] \to \R$. We consider here $T_s:[0,1]\to [0,1]$ the Manneville-Pomeau transformation. The autocorrelation function of the resulting process decays hyperbolically (or polynomially) and we obtain efficient methods to estimate the parameter s from a finite time series. As a consequence we also estimate the rate of convergence of the autocorrelation decay of these processes. We compare different estimation methods based on the periodogram function, on the smoothed periodogram function, on the variance of the partial sum and on the wavelet theory.

math.ST↗