The automatic Algorithm ARI developed in this paper can generate variates from a large class of unimodal discrete distributions. It is only necessary to know the mode of the distribution and to have a subprogram available that can evaluate the probabilities. In a set up step the algorithm constructs a table mountain shaped hat function. Then rejection inversion, a new variant of the rejection method for discrete distributions that needs only one uniform random number per iteration, is used to sample from the desired distribution. It is shown that the expeceted number of iterations is uniformly bounded for all T-concave discrete distributions. Utilizing a simple squeeze or an auxiliary table of moderate size, which is initialized during generation and not in the set up, Algorithm ARI is fast, at least as fast as the fastest known methods designed for the Poisson, binomial and hypergeometric distributions. The set up time of the algorithm is not affected by the size of the domain of the distribution and is about ten times longer than the generation of one variate. Compared with the very fast and well known alias and indexed search methods the set up of Algorithm ARI is much faster but the generation time is about two times slower. More important than the speed is the fact that Algorithm ARI is the first automatic algorithm that can generate samples from discrete distributions with heavy tails. (author's abstract) / Series: Preprint Series / Department of Applied Statistics and Data Processing

Στην εργασία αυτή παρουσιάζεται η πιθανογεννήτρια του αριθμού των απογόνων της ν-oστης γενιάς μια κλαδωτής ανέλιξης ως το πηλίκο των πιθανογεννήτριων δύο γεωμετρικών κατανομών. Στην βιβλιογραφία, με εξαίρεση δύο συγκεκριμένες περιπτώσεις (πηλίκα πιθανογεννητριών αρνητικής διωνυμικής με γεωμετρική, Kemp, 1979, και γεωμετρικής με Poisson Jayasree and Swamy, 2006), δεν έχει μελετηθεί το γενικότερο πρόβλημα των συνθηκών που επιτρέπουν το πηλίκο δύο πιθανογεννητριών να είναι η πιθανογεννήτρια μιας διακριτής μη αρνητικής τυχαίας μεταβλητής. Εδώ δίνονται οι ικανές και αναγκαίες συνθήκες για τα αντίστοιχα πηλίκα πιθανογεννητριών κατανομών από την οικογένεια Katz ή την οικογένεια Sundt and Jewell με την γεωμετρική κατανομή. Μελετάται επίσης και το πηλίκο απείρως διαιρετών κατανομών με την Poisson και παρουσιάζονται αναλυτικά τέτοια παραδείγματα. Διάφορες ιδιότητες των κατανομών που προκύπτουν εξετάζονται και γίνεται εκτίμηση των παραμέτρων. Στη συνέχεια, παρουσίαζεται μια διδιάστατη κλαδωτή ανέλιξη, δίνεται αναλυτικός τύπος για την πιθανογεννήτρια της από κοινού συνάρτησης κατανομής του πλήθους των δύο ειδών απογόνων της ν-oστης γενιάς, και αποδεικνύεται ότι αυτή μπορεί να γραφεί ως το πηλίκο των πιθανογεννήτριων δύο διδιαστάτων γεωμετρικών κατανομών. Μελετούμε γενικότερα το αντίστοιχο πρόβλημα για διδιάστατες τ.μ. και εξετάζουμε τις ικανές συνθήκες στις περιπτώσεις πηλίκου πιθανογεννητριών της διδιάστατης αρνητικής διωνυμικής με τη διδιάστατη γεωμετρική και της διδιάστατης αρνητικής διωνυμικής με τη διδιάστατη Poisson. Παρουσιάζονται αναγωγικές και αναλυτικές σχέσεις για τις πιθανότητες και τις παραγοντικές ροπές και μελετάται η μορφή των πιθανογεννητριών τόσο των περιθωρίων όσο και των δεσμευμένων κατανομών που προκύπτουν. / In this master thesis we observe, that the probability generating function of the number of the descendants of the n-th generation in a branching process, can be represented as the ratio of the probability generating functions (p.g.f.) of two geometric distributions. In the literature, with the exception of two particular cases (ratio of negative binomial with geometric, Kemp, 1979, and geometric with Poisson, Jayasree and Swamy, 2006), the general problem, for the conditions that allow the ratio of two p.g.f.’s to be the p.g.f. of a discrete non-negative random variable (r.v.), has not been considered. Here, are given the necessary and sufficient conditions for the ratios of the p.g.f. of a distribution from the Katz or the Sundt and Jewell family with the p.g.f. of a Geometric distribution. The ratio of an infinitely divisible r.v. with a Poisson r.v. is also studied and various such examples are presented in detail. Properties of these distributions are given and also parameters estimators are provided. In the sequel, a bivariate branching process is considered and the explicit form for the p.g.f. of the number of two type descendants in the n-th generation is derived. It is proved, that it can be written as the ratio of the p.g.f.’s of two bivariate geometric distributions. The sufficient conditions in the cases of the ratio of the bivariate negative binomial distribution with the bivariate geometric distribution and the bivariate negative binomial distribution with the bivariate Poisson distribution are examined. Recurrence relations and the explicit form of the probabilities and the factorial moments are given and the form of the p.g.f.’s for the marginals and the conditional distributions are studied.

As relações de dependência entre variáveis aleatórias é um dos assuntos mais discutidos em probabilidade e estatística e a forma mais abrangente de estudar essas relações é por meio da distribuição conjunta. Nos últimos anos vem crescendo a utilização de cópulas para representar a estrutura de dependência entre variáveis aleatórias em uma distribuição multivariada. Contudo, ainda existe pouca literatura sobre cópulas quando as distribuições marginais são discretas. No presente trabalho será apresentada uma proposta não-paramétrica de estimação da distribuição conjunta bivariada de variáveis aleatórias discretas utilizando cópulas e polinômios de Bernstein. / The relations of dependence between random variables is one of the most discussed topics in probability and statistics and the best way to study these relationships is through the joint distribution. In the last years has increased the use of copulas to represent the dependence structure among random variables in a multivariate distribution. However, there is still little literature on copulas when the marginal distributions are discrete. In this work we present a non-parametric approach for the estimation of the bivariate joint distribution of discrete random variables using copulas and Bernstein polynomials.

As relações de dependência entre variáveis aleatórias é um dos assuntos mais discutidos em probabilidade e estatística e a forma mais abrangente de estudar essas relações é por meio da distribuição conjunta. Nos últimos anos vem crescendo a utilização de cópulas para representar a estrutura de dependência entre variáveis aleatórias em uma distribuição multivariada. Contudo, ainda existe pouca literatura sobre cópulas quando as distribuições marginais são discretas. No presente trabalho será apresentada uma proposta não-paramétrica de estimação da distribuição conjunta bivariada de variáveis aleatórias discretas utilizando cópulas e polinômios de Bernstein. / The relations of dependence between random variables is one of the most discussed topics in probability and statistics and the best way to study these relationships is through the joint distribution. In the last years has increased the use of copulas to represent the dependence structure among random variables in a multivariate distribution. However, there is still little literature on copulas when the marginal distributions are discrete. In this work we present a non-parametric approach for the estimation of the bivariate joint distribution of discrete random variables using copulas and Bernstein polynomials.

Generalized autoregressive moving average (GARMA) models are a class of models that was developed for extending the univariate Gaussian ARMA time series model to a flexible observation-driven model for non-Gaussian time series data. This work presents the GARMA model with discrete distributions and application of resampling techniques to this class of models. We also proposed The Bayesian approach on GARMA models. The TGARMA (Transformed Generalized Autoregressive Moving Average) models was proposed, using the Box-Cox power transformation. Last but not least we proposed the Bayesian approach for the TGARMA (Transformed Generalized Autoregressive Moving Average). / Modelos Autoregressivos e de médias móveis generalizados (GARMA) são uma classe de modelos que foi desenvolvida para extender os conhecidos modelos ARMA com distribuição Gaussiana para um cenário de series temporais não Gaussianas. Este trabalho apresenta os modelos GARMA aplicados a distribuições discretas, e alguns métodos de reamostragem aplicados neste contexto. É proposto neste trabalho uma abordagem Bayesiana para os modelos GARMA. O trabalho da continuidade apresentando os modelos GARMA transformados, utilizando a transformação de Box-Cox. E por último porém não menos importante uma abordagem Bayesiana para os modelos GARMA transformados.

We use inequalities to design short universal algorithms that can be used to generate random variates from large classes of univariate continuous or discrete distributions (including all log-concave distributions). The expected time is uniformly bounded over all these distributions. The algorithms can be implemented in a few lines of high level language code. In opposition to other black-box algorithms hardly any setup step is required and thus it is superior in the changing parameter case. (author's abstract) / Series: Preprint Series / Department of Applied Statistics and Data Processing

We use inequalities to design short universal algorithms that can be used to generate random variates from large classes of univariate continuous or discrete distributions (including all log-concave distributions). The expected time is uniformly bounded over all these distributions for a particular generator. The algorithms can be implemented in a few lines of high level language code. (author's abstract) / Series: Preprint Series / Department of Applied Statistics and Data Processing

Mελετάται η μίξη δύο συνεχών κατανομών με αύξοντα ρυθμό αποτυχίας και δίνονται συνθήκες για να έχει η μίξη φθίνοντα ρυθμό αποτυχίας. Όταν η μία από τις δύο κατανομές της μίξης είναι η εκθετική γίνεται αντιστροφή του ρυθμού αποτυχίας. Στην περίπτωση της μίξης δύο κανονικών κατανομών παρουσιάζεται ο τρόπος που συνδέεται το πλήθος των κορυφών της πυκνότητας με τον ρυθμό αποτυχίας της μίξης. Mελετάται επίσης, η μονοτονία του ρυθμού αποτυχίας διακριτών κατανομών χρησιμοποιώντας τον λόγο δύο διαδοχικών πιθανοτήτων και δίδεται μία συνθήκη για να έχει η μίξη δύο διακριτών κατανομών φθίνοντα ρυθμό αποτυχίας όταν η μία από τις δύο κατανομές της μίξης είναι η γεωμετρική. Τέλος, χρησιμοποιώντας τον λόγο διαδοχικών πιθανοτήτων, μελετούμε την μονοτονία του ρυθμού αποτυχίας για διδιάστατες διακριτές κατανομές. / The mixture of two continuous distributions, with increasing failure rates, is considered and the necessary conditions to have decreasing failure rate (DFR) are given. When one of these distributions is the Exponential, reversal of the failure rate is observed. In the case of two normal distributions the failure rate is associated with the number of modes. It is also considered the failure rate for discrete distributions in regard with the ratio of two consecutive probabilities. A condition to have DFR is given when one of the distributions of the mixture is the geometric. Finally, we make use of the ratio of two consecutive probabilities to study the failure rate for bivariate discrete distributions.

Submitted by Aelson Maciera (aelsoncm@terra.com.br) on 2017-08-03T20:04:27Z No. of bitstreams: 1 TeseBSA.pdf: 10322083 bytes, checksum: 4c30c490934f23dbad9d5a1f087ef182 (MD5) / Approved for entry into archive by Ronildo Prado (ronisp@ufscar.br) on 2017-08-08T19:09:23Z (GMT) No. of bitstreams: 1 TeseBSA.pdf: 10322083 bytes, checksum: 4c30c490934f23dbad9d5a1f087ef182 (MD5) / Approved for entry into archive by Ronildo Prado (ronisp@ufscar.br) on 2017-08-08T19:09:30Z (GMT) No. of bitstreams: 1 TeseBSA.pdf: 10322083 bytes, checksum: 4c30c490934f23dbad9d5a1f087ef182 (MD5) / Made available in DSpace on 2017-08-08T19:15:39Z (GMT). No. of bitstreams: 1 TeseBSA.pdf: 10322083 bytes, checksum: 4c30c490934f23dbad9d5a1f087ef182 (MD5) Previous issue date: 2016-12-16 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Generalized autoregressive moving average (GARMA) models are a class of models that was developed for extending the univariate Gaussian ARMA time series model to a flexible observation-driven model for non-Gaussian time series data. This work presents the GARMA model with discrete distributions and application of resampling techniques to this class of models. We also proposed The Bayesian approach on GARMA models. The TGARMA (Transformed Generalized Autoregressive Moving Average) models was proposed, using the Box-Cox power transformation. Last but not least we proposed the Bayesian approach for the TGARMA (Transformed Generalized Autoregressive Moving Average). / Modelos Autoregressivos e de médias móveis generalizados (GARMA) são uma classe de modelos que foi desenvolvida para extender os conhecidos modelos ARMA com distribuição Gaussiana para um cenário de series temporais não Gaussianas. Este trabalho apresenta os modelos GARMA aplicados a distribuições discretas, e alguns métodos de reamostragem aplicados neste contexto. É proposto neste trabalho uma abordagem Bayesiana para os modelos GARMA. O trabalho da continuidade apresentando os modelos GARMA transformados, utilizando a transformação de Box-Cox. E por último porém não menos importante uma abordagem Bayesiana para os modelos GARMA transformados.

This thesis contains theory of probability and statistical sets. Solved and unsolved problems of probability, random variable and distributions random variable, random vector, statistical sets, regression and correlation analysis. Unsolved problems contains solutions.