Search

Subject and purpose. The task of designing various radio engineering devices, such as filters, delay lines, antennas with a given radiation pattern, requires the development of methods for generating random sequences (the values of the system parameters) with given correlation properties, since the spectral characteristics of the listed and similar to them systems are expressed in terms of the Fourier transforms of correlators. The purpose of this paper is to represent the function of the transition probability of random sequences with long-range correlations in a form convenient for numerical generation of sequences, and to study the statistical properties of the latter. Method and methodology. An adequate mathematical tool for solving such problems is the higher order Markov chains. The statistical characteristics of these objects are determined by their transition probability function, which in the general case can have a very complex form. In this paper, the transition probability function is assumed to be additive and linear with respect to the values of the random variable. It is assumed that the state space of the sequence belongs to the set of real numbers. Results. The equations that relate the correlation functions of the sequence to the weight coefficients of the memory function, determined in their turn by the transition probability function, are derived and analytically solved. Conclusion. It is shown that the correlation functions of the additive Markov chain are completely determined by the variance of the random variable and the weight coefficients of the memory function. The agreement of the obtained analytical results with the results of numerical realization of the additive Markov sequence is demonstrated. Examples of possible correlation scenarios in higher order additive linear chains are given.

The Markov sequence problem aims at the description of possible eigenvalues of symmetric Markov operators with some given orthonormal basis as eigenvector decomposition. A fundamental tool for their description is the hypergroup property. We first present the general Markov sequence problem and provide the classical examples, most of them associated with the classical families of orthogonal polynomials. We then concentrate on the hypergroup property, and provide a general method to obtain it, inspired by a fundamental work of Carlen, Geronimo and Loss. Using this technique and a few properties of diffusion operators having polynomial eigenvectors, we then provide a simplified proof of the hypergroup property for the Jacobi polynomials (Gasper’s theorem) on the unit interval. We finally investigate various generalizations of this property for the family of Dirichlet laws on the simplex.

We consider the problem of finite-dimensional parameter estimation for Markov sequences in the asymptotic of large samples. The asymptotic behavior of the Bayesian estimators and the estimators of the method of moments are described. It is shown that these estimators are consistent and asymptotically normal. We show that the Bayesian estimator is asymptotically efficient. We propose a particular non-linear autoregressive process as example and we illustrate the properties of these estimators with the help of numerical simulations.

In the first part of this dissertation we consider several problems of finite-dimensional parameter estimation for Markov sequences in the asymptotics of large samples. The asymptotic behavior of the Bayesian estimators and the estimators of the method of moments are described. It is shown that under regularity conditions these estimators are consistent and asymptotically normal. We show that the Bayesian estimator is asymptotically efficient. The one-step and two-step maximum likelihood estimator-processes are studied. These estimators allow us to construct the asymptotically efficient estimators based on some preliminary estimators, say, the estimators of the method of moments or Bayes estimator and the one-step maximum likelihood estimator structure. We propose particular non-linear autoregressive processes as examples and we illustrate the properties of these estimators with the help of numerical simulations. In the second part we give theapplications of Markov processes in health economics. We compare homogeneous and non-homogeneous Markov models for cost-effectiveness analysis of routine use of transparent dressings containing a chlorhexidine gluconate gel pad versus standard transparent dressings. The antimicrobial dressing protects central vascular accesses reducing the risk of catheter-related bloodstream infections. The impact of the modeling approach on the decision of adopting antimicrobialdressings for critically-ill patients is discussed., Dans la première partie de cette thèse, nous considérons plusieurs problèmes d'estimation de paramètre de dimension finie pour les séquences de Markov dans l'asymptotique des grands échantillons. Le comportement asymptotique des estimateurs bayésiens et les estimateurs obtenus par la méthode des moments sont décrits. Nous montrons que sous les conditions de régularité ces estimateurs sont consistants et asymptotiquement normaux et que l'estimateur bayésien est asymptotiquement efficace. Les estimateur-processus du maximum de vraisemblance un-pas et deux-pas sont étudiés. Ces estimateurs nous permettent de construire des estimateurs asymptotiquement efficaces sur la base de certainsestimateurs préliminaires, par exemple, les estimateurs obtenus par la méthode des moments ou l'estimateur deBayes et la structure de l'estimateur du maximum de vraisemblance un-pas. Nous proposons notamment des processus autorégressifs non linéaires comme exemple et nous illustrons les propriétés de ces estimateurs à l'aide de simulations numériques. Dans la deuxième partie, nous donnons les applications de processus de Markov en économie de la santé. Nous comparons les modèles de Markov homogènes et non-homogènes pour l'analyse coût-efficacité de l'utilisation depansements transparents contenant un gel de gluconate de chlorhexidine par rapport aux pansements transparents standard. Le pansement antimicrobien protège les accès vasculaire centrale et réduit le risque de bactériémies liées aux cathéters. L'impact de l'approche de modélisation sur la décision d'adopter des pansements antimicrobiens pour les patients gravement malades est discuté.

In the first part of this dissertation we consider several problems of finite-dimensional parameter estimation for Markov sequences in the asymptotics of large samples. The asymptotic behavior of the Bayesian estimators and the estimators of the method of moments are described. It is shown that under regularity conditions these estimators are consistent and asymptotically normal. We show that the Bayesian estimator is asymptotically efficient. The one-step and two-step maximum likelihood estimator-processes are studied. These estimators allow us to construct the asymptotically efficient estimators based on some preliminary estimators, say, the estimators of the method of moments or Bayes estimator and the one-step maximum likelihood estimator structure. We propose particular non-linear autoregressive processes as examples and we illustrate the properties of these estimators with the help of numerical simulations. In the second part we give theapplications of Markov processes in health economics. We compare homogeneous and non-homogeneous Markov models for cost-effectiveness analysis of routine use of transparent dressings containing a chlorhexidine gluconate gel pad versus standard transparent dressings. The antimicrobial dressing protects central vascular accesses reducing the risk of catheter-related bloodstream infections. The impact of the modeling approach on the decision of adopting antimicrobialdressings for critically-ill patients is discussed., Dans la première partie de cette thèse, nous considérons plusieurs problèmes d'estimation de paramètre de dimension finie pour les séquences de Markov dans l'asymptotique des grands échantillons. Le comportement asymptotique des estimateurs bayésiens et les estimateurs obtenus par la méthode des moments sont décrits. Nous montrons que sous les conditions de régularité ces estimateurs sont consistants et asymptotiquement normaux et que l'estimateur bayésien est asymptotiquement efficace. Les estimateur-processus du maximum de vraisemblance un-pas et deux-pas sont étudiés. Ces estimateurs nous permettent de construire des estimateurs asymptotiquement efficaces sur la base de certainsestimateurs préliminaires, par exemple, les estimateurs obtenus par la méthode des moments ou l'estimateur deBayes et la structure de l'estimateur du maximum de vraisemblance un-pas. Nous proposons notamment des processus autorégressifs non linéaires comme exemple et nous illustrons les propriétés de ces estimateurs à l'aide de simulations numériques. Dans la deuxième partie, nous donnons les applications de processus de Markov en économie de la santé. Nous comparons les modèles de Markov homogènes et non-homogènes pour l'analyse coût-efficacité de l'utilisation depansements transparents contenant un gel de gluconate de chlorhexidine par rapport aux pansements transparents standard. Le pansement antimicrobien protège les accès vasculaire centrale et réduit le risque de bactériémies liées aux cathéters. L'impact de l'approche de modélisation sur la décision d'adopter des pansements antimicrobiens pour les patients gravement malades est discuté.