publications

Scientific papers

2024

1. Anomaly Detection based on Markov Data: A Statistical Depth Approach

   Carlos A. Fernández, and Stephan Clémençon

   2024

   Submitted to SDM 2025

   Abs arXiv

   It is the main purpose of this article to extend the notion of statistical depth to the case of sample paths of a Markov chain, a very popular probabilistic model to describe parsimoniously random phenomena with a temporal causality. Initially introduced to define a center-outward ordering of points in the support of a multivariate distribution, depth functions permit to generalize the notions of quantiles and (signed) ranks for observations in \mathbbR^d with d>1, as well as statistical procedures based on such quantities, for (unsupervised) anomaly detection tasks in particular. In this paper, overcoming the lack of natural order on the torus composed of all possible trajectories of finite length, we develop a general theoretical framework for evaluating the depth of a Markov sample path and recovering it statistically from an estimate of its transition probability with (non-) asymptotic guarantees. We also detail its numerous applications, focusing particularly on anomaly detection, a key task in various fields involving the analysis of (supposedly) Markov time-series (\textite.g. health monitoring of complex infrastructures, security). Beyond the description of the methodology promoted and the statistical analysis carried out to guarantee its validity, numerical experiments are displayed, providing strong empirical evidence of the relevance of the novel concept we introduce here to quantify the degree of abnormality of Markov path sequences of variable length.

2. Regenerative bootstrap for β-null recurrent Markov chains

   Carlos A. Fernández

   Electron. J. Statist., 2024

   Abs DOI arXiv

   Two regeneration-based bootstrap methods, namely, the \textitRegeneration based-bootstrap \citeAthreyaFuh1992, Somnat-1993 and the \textitRegenerative Block bootstrap \citeBertail2006 are shown to be valid for the problem of estimating the integral of a function with respect to the invariant measure in a β-null recurrent Markov chain with an accessible atom. An extension of the Central Limit Theorem for randomly indexed sequences is also presented.

3. Tail Index Estimation for Discrete Heavy-Tailed Distributions with Application to Statistical Inference for Regular Markov Chains

   Patrice Bertail, Stephan Clémençon, and Carlos A. Fernández

   2024

   Submitted to TEST

   Abs arXiv

   It is the purpose of this paper to investigate the issue of estimating the regularity index \(β>0\)of a discrete heavy-tailed r.v. S, \textiti.e. a r.v. S valued in \mathbbN^* such that \mathbbP(S>n)=L(n)⋅n^-β for all n≥1, where L:\mathbbR^*_+\to \mathbbR_+ is a slowly varying function. Such discrete probability laws, referred to as generalized Zipf’s laws sometimes, are commonly used to model rank-size distributions after a preliminary range segmentation in a wide variety of areas such as \textite.g. quantitative linguistics, social sciences or information theory. As a first go, we consider the situation where inference is based on independent copies S_1,; \ldots,; S_n of the generic variable S. Just like the popular Hill estimator in the continuous heavy-tail situation, the estimator \widehatβ we propose can be derived by means of a suitable reformulation of the regularly varying condition, replacing S’s survivor function by its empirical counterpart. Under mild assumptions, a non-asymptotic bound for the deviation between \widehatβ and βis established, as well as limit results (consistency and asymptotic normality). Beyond the i.i.d. case, the inference method proposed is extended to the estimation of the regularity index of a regenerative β-null recurrent Markov chain. Since the parameter βcan be then viewed as the tail index of the (regularly varying) distribution of the return time of the chain X to any (pseudo-) regenerative set, in this case, the estimator is constructed from the successive regeneration times. Because the durations between consecutive regeneration times are asymptotically independent, we can prove that the consistency of the estimator promoted is preserved. In addition to the theoretical analysis carried out, simulation results provide empirical evidence of the relevance of the inference technique proposed.

4. Harris Recurrent Markov Chains and Nonlinear Monotone Cointegrated Models

   Patrice Bertail, Cécile Durot, and Carlos A. Fernández

   2024

   Preprint

   Abs arXiv

   In this paper, we study a nonlinear cointegration-type model of the form \(Z_t = f_0(X_t) + W_t\)where \(f_0\)is a monotone function and \(X_t\)is a Harris recurrent Markov chain. We use a nonparametric Least Square Estimator to locally estimate \(f_0\), and under mild conditions, we show its strong consistency and obtain its rate of convergence. New results (of the Glivenko-Cantelli type) for localized null recurrent Markov chains are also proved.

2019

1. A variant of the Geo/G/1 queues with disasters and general repair times

   Yoel G. Yera, Carlos A. Fernández, and José E. Valdés

   Communications in Statistics - Theory and Methods, 2019

   Abs DOI Bib

   This paper deals with Geo/G/1 queues with a repairable server. The server is subject to failure due to a disaster arrival, which can occur while the server is turned on and not only when it is busy. At a failure instant, the server is turned off and its repair process begins. During the repair process, two models are considered. For both models, we present the PGF and the expected number of clients in the system in the steady state.

   @article{doi:10.1080/03610926.2018.1528368,
     author = {Yera, Yoel G. and Fern{\'a}ndez, Carlos A. and Vald{\'e}s, Jos{\'e} E.},
     doi = {10.1080/03610926.2018.1528368},
     eprint = {https://doi.org/10.1080/03610926.2018.1528368},
     journal = {Communications in Statistics - Theory and Methods},
     number = {24},
     pages = {6119--6133},
     publisher = {Taylor \& Francis},
     title = {A variant of the Geo/G/1 queues with disasters and general repair times},
     url = {https://doi.org/10.1080/03610926.2018.1528368},
     volume = {48},
     year = {2019},
     bdsk-url-1 = {https://doi.org/10.1080/03610926.2018.1528368},
   }

Theses

2023

1. Estimation problems on null recurrent time series

   Carlos A. Fernández

   Université Paris Nanterre PhD thesis , 2023

   Abs Bib HTML PDF

   In the field of Markov chain theory, β-null recurrent Markov chains represent a class of stochastic processes that exhibit challenging and peculiar properties. These nonstationary chains possess infinite invariant measures, making the estimation problems associated with them particularly intricate.This thesis delves into several estimation problems in the context of β-null recurrent Markov chains, providing new insights and methodologies to tackle these challenges. Our first contribution is the proposal of a tail index estimator for generalized discrete Pareto distributions, which is then used to estimate the parameter β in atomic β-null recurrent Markov chains. The second contribution involves the adaptation and validation of the Regeneration-based bootstrap and Regenerative Block bootstrap methods for these types of chains. Lastly, we develop an estimator for monotone functions in nonlinear cointegrated models, where the underlying process is a Harris recurrent Markov chain (positive or β-null recurrent).

   @phdthesis{sanz2023estimation,
     title = {Estimation problems on null recurrent time series},
     author = {Fernández, Carlos A.},
     year = {2023},
     school = {Université Paris Nanterre},
     url = {https://bdr.parisnanterre.fr/theses/internet/2023/2023PA100040/2023PA100040.pdf},
   }

2016

1. On the number of non-trivial solutions of diagonal equaitons over finite fields

   Carlos A. Fernández

   Universidad de La Habana Master thesis (in Spanish) , 2016

   Abs Bib HTML PDF

   Two new exponential sums are presented in this investigation. This new sums are used to proof a theorem that allows to calculate the number of non trivial solutions of a diagonal equation over a finite field. Using this theorem, a new proof of Fermat Last Theorem over Finite Fields is presented.

   @mastersthesis{sanz2016cantidad,
     title = {On the number of non-trivial solutions of diagonal equaitons over finite fields},
     author = {Fern{\'a}ndez, Carlos A.},
     year = {2016},
     school = {Universidad de La Habana},
     url = {https://www.researchgate.net/profile/Carlos-Fernandez-Sanz/publication/369857848_Cantidad_de_soluciones_no_triviales_de_ecuaciones_diagonales_sobre_cuerpos_finitos/links/642fd9dfad9b6d17dc3f205b/Cantidad-de-soluciones-no-triviales-de-ecuaciones-diagonales-sobre-cuerpos-finitos.pdf},
   }

2013

1. Una visión algebraica de las sumas de Gauss y de Jacobi

   Carlos A. Fernández

   Universidad de La Habana License thesis (in Spanish) , 2013

   Abs Bib HTML PDF

   This investigation will survey two types of character sums: Gauss sums and Jacobi sums. Along the way, the theory developed will be used to give a sufficient condition for the solubility of diagonal equations over finite fields. Also, will be shown new proofs of known theorems, just like some applications and generalizations.

   @mastersthesis{sanz2013cantidad,
     title = {Una visión algebraica de las sumas de Gauss y de Jacobi},
     author = {Fern{\'a}ndez, Carlos A.},
     year = {2013},
     school = {Universidad de La Habana},
     url = {https://www.researchgate.net/profile/Carlos-Fernandez-Sanz/publication/259801416_Tesis_de_licenciatura_Una_vision_algebraica_de_las_Sumas_de_Gauss_y_de_Jacobi_Gauss_and_Jacobi_sums_from_an_algebraic_point_of_view/links/00b4952dec2e1a31d7000000/Tesis-de-licenciatura-Una-vision-algebraica-de-las-Sumas-de-Gauss-y-de-Jacobi-Gauss-and-Jacobi-sums-from-an-algebraic-point-of-view.pdf},
   }