The models for volatility, autoregressive conditional heteroscedastic (ARCH) and generalized autoregressive conditional heteroscedastic (GARCH) are discussed. Stationarity condition and forecasting for simple ARCH(1) and GARCH(1,1) models are given. The model for discrete time series is proposed to be negative binomial integer-valued GARCH model, which is a generalization of the Poisson INGARCH model. The stationarity conditions and the autocorrelation function are given. For parameter estimation, three methodologies are presented with a focus on maximum likelihood approach. Simulation study on a sample size of 100 and 500 are carried out and the results are presented. An application of the model to a real time series with numerical example is given indicating that the proposed methodology performs better than the Poisson and double Poisson model-based methods.

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An integer-valued polynomial is a polynomial with rational coefficients that takes an integer value when evaluated at an integer. The binomial polynomials form a regular basis for the Z-module of all integer-valued polynomials. Using the idea of a p-ordering and a p-sequence, Bhargava describes a similar characterization for polynomials that are integer-valued on some subset of the integers. This thesis focuses on characterizing the polynomials that are integer-valued on the Fibonacci numbers. For a certain class of primes p, we give a formula for the p-sequence of the Fibonacci numbers and an algorithm for finding a p-ordering using Coelho and Parry’s results on the distribution of the Fibonacci numbers modulo powers of primes. Knowing the p-sequence, we can then find a p-local regular basis for the polynomials that are integer-valued on the Fibonacci numbers using Bhargava’s methods. A regular basis can be constructed from p-local bases for all primes p.

This dissertation proposes new statistical methods for modeling and inference for two specific types of time series: integer-valued data and multivariate nonstationary extreme data. We rely on the class of integer-valued autoregressive (INAR) processes for the former, proposing a novel, flexible and elegant way of modeling count phenomena. As for the latter, we are interested in the human brain and its multi-channel electroencephalogram (EEG) recordings, a natural source of extreme events. Thus, we develop new extreme value theory methods for analyzing such data, whether in modeling the conditional extremal dependence for brain connectivity or clustering extreme brain communities of EEG channels. Regarding integer-valued time series, INAR processes are generally defined by specifying the thinning operator and either the innovations or the marginal distributions. The major limitations of such processes include difficulties deriving the marginal properties and justifying the choice of the thinning operator. To overcome these drawbacks, this dissertation proposes a novel approach for building an INAR model that offers the flexibility to prespecify both marginal and innovation distributions. Thus, the thinning operator is no longer subjectively selected but is rather a direct consequence of the marginal and innovation distributions specified by the modeler. Novel INAR processes are introduced following this perspective; these processes include a model with geometric marginal and innovation distributions (Geo-INAR) and models with bounded innovations. We explore the Geo-INAR model, which is a natural alternative to the classical Poisson INAR model. The Geo-INAR process has interesting stochastic properties, such as MA($\infty$) representation, time reversibility, and closed forms for the $h$-th-order transition probabilities, which enables a natural framework to perform coherent forecasting. In the front of multivariate nonstationary extreme data, the focus lies on multi-channel epilepsy data. Epilepsy is a chronic neurological disorder affecting more than 50 million people globally. An epileptic seizure acts like a temporary shock to the neuronal system, disrupting normal electrical activity in the brain. Epilepsy is frequently diagnosed with EEGs. Current statistical approaches for analyzing EEGs use spectral and coherence analysis, which do not focus on extreme behavior in EEGs (such as bursts in amplitude), neglecting that neuronal oscillations exhibit non-Gaussian heavy-tailed probability distributions. To overcome this limitation, this dissertation proposes new approaches to characterize brain connectivity based on extremal features of EEG signals. Two extreme-valued methods to study alterations in the brain network are proposed. One method is Conex-Connect, a pioneering approach linking the extreme amplitudes of a reference EEG channel with the other channels in the brain network. The other method is Club Exco, which clusters multi-channel EEG data based on a spherical $k$-means procedure applied to the "pseudo-angles," derived from extreme amplitudes of EEG signals. Both methods provide new insights into how the brain network organizes itself during an extreme event, such as an epileptic seizure, in contrast to a baseline state.

Title: Models of integer-valued time series with random coefficients Author: Petra Burdejová Department: Department of Probability and Mathematical Statistics Supervisor: Doc. RNDr. Zuzana Prášková, CSc. Abstract: In the presented thesis, a generalized integer-valued autoregres- sive process of the order p (GINAR(p)) is considered first. The main aim is taken to introduction of random coefficient integer-valued autoregressive process (RCINAR(p)). We use a thinning operator in order to define the processes. The main characteristics of GINAR(p) and RCINAR(p) are obtained. Condi- tions for stationarity and ergodicity are stated. Three methods of estimation (Yule-Walker, Conditional least squares, Generalized method of moments) are given and compared in simulation with respect to the mean squared error (MSE). At the end, RCINAR(3) model is applied to a real dataset representing a number of earthquakes per year. Keywords: thinning operator, random coefficients, integer-valued time se- ries, GINAR, RCINAR

Les processus autorégressifs à bifurcation (BAR) ont été au centre de nombreux travaux de recherche ces dernières années. Ces processus, qui sont l'adaptation à un arbre binaire des processus autorégressifs, sont en effet d'intérêt en biologie puisque la structure de l'arbre binaire permet une analogie aisée avec la division cellulaire. L'objectif de cette thèse est l'estimation les paramètres de variantes de ces processus autorégressifs à bifurcation, à savoir les processus BAR à valeurs entières et les processus BAR à coefficients aléatoires. Dans un premier temps, nous nous intéressons aux processus BAR à valeurs entières. Nous établissons, via une approche martingale, la convergence presque sûre des estimateurs des moindres carrés pondérés considérés, ainsi qu'une vitesse de convergence de ces estimateurs, une loi forte quadratique et leur comportement asymptotiquement normal. Dans un second temps, on étudie les processus BAR à coefficients aléatoires. Cette étude permet d'étendre le concept de processus autorégressifs à bifurcation en généralisant le côté aléatoire de l'évolution. Nous établissons les mêmes résultats asymptotiques que pour la première étude. Enfin, nous concluons cette thèse par une autre approche des processus BAR à coefficients aléatoires où l'on ne pondère plus nos estimateurs des moindres carrés en tirant parti du théorème de Rademacher-Menchov. / Bifurcating autoregressive (BAR) processes have been widely investigated this past few years. Those processes, which are an adjustment of autoregressive processes to a binary tree structure, are indeed of interest concerning biology since the binary tree structure allows an easy analogy with cell division. The aim of this thesis is to estimate the parameters of some variations of those BAR processes, namely the integer-valued BAR processes and the random coefficients BAR processes. First, we will have a look to integer-valued BAR processes. We establish, via a martingale approach, the almost sure convergence of the weighted least squares estimators of interest, together with a rate of convergence, a quadratic strong law and their asymptotic normality. Secondly, we study the random coefficients BAR processes. The study allows to extend the principle of bifurcating autoregressive processes by enlarging the randomness of the evolution. We establish the same asymptotic results as for the first study. Finally, we conclude this thesis with an other approach of random coefficient BAR processes where we do not weight our least squares estimators any more by making good use of the Rademacher-Menchov theorem.

Dans cette thèse, nous étudions des modèles de moyennes conditionnelles de séries temporelles à valeurs entières. Tout d’abord, nous proposons l’estimateur de quasi maximum de vraisemblance de Poisson (EQMVP) pour les paramètres de la moyenne conditionnelle. Nous montrons que, sous des conditions générales de régularité, cet estimateur est consistant et asymptotiquement normal pour une grande classe de modèles. Étant donné que les paramètres de la moyenne conditionnelle de certains modèles sont positivement contraints, comme par exemple dans les modèles INAR (INteger-valued AutoRegressive) et les modèles INGARCH (INteger-valued Generalized AutoRegressive Conditional Heteroscedastic), nous étudions la distribution asymptotique de l’EQMVP lorsque le paramètre est sur le bord de l’espace des paramètres. En tenant compte de cette dernière situation, nous déduisons deux versions modifiées du test de Wald pour la significativité des paramètres et pour la moyenne conditionnelle constante. Par la suite, nous accordons une attention particulière au problème de validation des modèles des séries temporelles à valeurs entières en proposant un test portmanteau pour l’adéquation de l’ajustement. Nous dérivons la distribution jointe de l’EQMVP et des autocovariances résiduelles empiriques. Puis, nous déduisons la distribution asymptotique des autocovariances résiduelles estimées, et aussi la statistique du test. Enfin, nous proposons l’EQMVP pour estimer équation-par-équation (EpE) les paramètres de la moyenne conditionnelle des séries temporelles multivariées à valeurs entières. Nous présentons les hypothèses de régularité sous lesquelles l’EQMVP-EpE est consistant et asymptotiquement normal, et appliquons les résultats obtenus à plusieurs modèles des séries temporelles multivariées à valeurs entières. / The framework of this PhD dissertation is the conditional mean count time seriesmodels. We propose the Poisson quasi-maximum likelihood estimator (PQMLE) for the conditional mean parameters. We show that, under quite general regularityconditions, this estimator is consistent and asymptotically normal for a wide classeof count time series models. Since the conditional mean parameters of some modelsare positively constrained, as, for example, in the integer-valued autoregressive (INAR) and in the integer-valued generalized autoregressive conditional heteroscedasticity (INGARCH), we study the asymptotic distribution of this estimator when the parameter lies at the boundary of the parameter space. We deduce a Waldtype test for the significance of the parameters and another Wald-type test for the constance of the conditional mean. Subsequently, we propose a robust and general goodness-of-fit test for the count time series models. We derive the joint distribution of the PQMLE and of the empirical residual autocovariances. Then, we deduce the asymptotic distribution of the estimated residual autocovariances and also of a portmanteau test. Finally, we propose the PQMLE for estimating, equation-by-equation (EbE), the conditional mean parameters of a multivariate time series of counts. By using slightly different assumptions from those given for PQMLE, we show the consistency and the asymptotic normality of this estimator for a considerable variety of multivariate count time series models.