Queueing theory has often been applied to study communication and service queueing systems such as call centers, hospital emergency departments and ride-sharing platforms. Unfortunately, it is complicated to analyze queueing systems. That is largely because the arrival and service processes that mainly determine a queueing system are uncertain and must be represented as stochastic processes that are difficult to analyze. In response, service providers might be able to partially capture the main characteristics of systems given partial data information and limited domain knowledge. An effective engineering response is to develop tractable approximations to approximate queueing characteristics of interest that depend on critical partial information. In this thesis, we contribute to developing high-quality approximations by studying tight bounds for the transient and the steady-state mean waiting time given partial information. We focus on single-server queues and multi-server queues with the unlimited waiting room, the first-come-first-served service discipline, and independent sequences of independent and identically distributed sequences of interarrival times and service times. We assume some partial information is known, e.g., the first two moments of inter-arrival and service time distributions. For the single-server GI/GI/1 model, we first study the tight upper bounds for the mean and higher moments of the steady-state waiting time given the first two moments of the inter-arrival time and service-time distributions. We apply the theory of Tchebycheff systems to obtain sufficient conditions for classical two-point distributions to yield the extreme values. For the tight upper bound of the transient mean waiting time, we formulate the problem as a non-convex non-linear program, derive the gradient of the transient mean waiting time over distributions with finite support, and apply classical non-linear programming theory to characterize stationary points. We then develop and apply a stochastic variant of the conditional gradient algorithm to find a stationary point for any given service-time distribution. We also establish necessary conditions and sufficient conditions for stationary points to be three-point distributions or special two-point distributions. Our studies indicate that the tight upper bound for the steady-state mean waiting time is attained asymptotically by two-point distributions as the upper mass point of the service-time distribution increases and the probability decreases, while one mass of the inter-arrival time distribution is fixed at 0. We then develop effective numerical and simulation algorithms to compute the tight upper bound. The algorithms are aided by reductions of the special queues with extremal inter-arrival time and extremal service-time distributions to D/GI/1 and GI/D/1 models. Combining these reductions yields an overall representation in terms of a D/RS(D)/1 discrete-time model involving a geometric random sum of deterministic random variables, where the two deterministic random variables have different values, so that the extremal waiting times need not have a lattice distribution. We finally evaluate the tight upper bound to show that it offers a significant improvement over established bounds. In order to understand queueing performance given only partial information, we propose determining intervals of likely performance measures given that limited information. We illustrate this approach for the steady-state waiting time distribution in the GI/GI/K queue given the first two moments of the inter-arrival time and service time distributions plus additional information about these underlying distributions, including support bounds, higher moments, and Laplace transform values. As a theoretical basis, we apply the theory of Tchebycheff systems to determine extremal models (yielding tight upper and lower bounds) on the asymptotic decay rate of the steady-state waiting-time tail probability, as in the Kingman-Lundberg bound and large deviations asymptotics. We then can use these extremal models to indicate likely intervals of other performance measures. We illustrate by constructing such intervals of likely mean waiting times. Without extra information, the extremal models involve two-point distributions, which yield a wide range for the mean. Adding constraints on the third moment and a transform value produces three-point extremal distributions, which significantly reduce the range, yielding practical levels of accuracy.

Em analise de dados que apresentam certo grau de assimetria a suposicao que as observações seguem uma distribuição normal, pode resultar ser uma suposição irreal e a aplicação deste modelo pode ocultar características importantes do modelo verdadeiro. Este tipo de situação deu forca á aplicação de modelo assimétricos, destacando-se entre estes a família de distribuições skew-symmetric, desenvolvida por Azzalini (1985). Neste trabalho nos apresentamos uma segunda proposta para a anàlise de dados com presença importante de assimetria e/ou curtose, comparado com a distribuição normal. Nós apresentamos e estudamos algumas propriedades dos modelos alfa-potência e log-alfa-potência, onde também estudamos o problema de estimação, as matrizes de informação observada e esperada de Fisher e o grau do viés dos estimadores mediante alguns processos de simulação. Nós introduzimos um modelo mais estável que o modelo alfa- potência do qual derivamos o caso bimodal desta distribuição e introduzimos os modelos bimodal simêtrico e assimêtrico alfa-potencia. Posteriormente nós estendemos a distribuição alfa-potência para o caso do modelo Birnbaum-Saunders, estudamos as propriedades deste novo modelo, desenvolvemos estimadores para os parametros e propomos estimadores com viés corrigido. Também introduzimos o modelo de regressão alfa-potência para dados censurados e não censurados e para o modelo de regressão log-linear Birnbaum-Saunders; aqui nós derivamos os estimadores dos parâmetros e estudamos algumas técnicas de validação dos modelos. Por ultimo nós fazemos a extensão multivariada do modelo alfa-potência e estudamos alguns processos de estimação dos parâmetros. Para todos os casos estudados apresentam-se ilustrações com dados já analisados previamente com outras suposições de distribuições. / In data analysis where data present certain degree of asymmetry the assunption of normality can result in an unreal situation and the application of this model can hide important caracteristics of the true model. Situations of this type has given strength to the use of asymmetric models with special emphasis on the skew-symmetric distribution developed by Azzalini (1985). In this work we present an alternative for data analysis in the presence of signi¯cant asymmetry or kurtosis, when compared with the normal distribution, as well as other situations that involve such model. We present and study of the properties of the ®-power and log-®-power distributions, where we also study the estimation problem, the observed and expected information matrices and the degree of bias in estimation using simulation procedures. A °exible model version is proposed for the ®-power distribution, following an extension to a bimodal version. Follows next an extension of the Birnbaum-Saunders distribution using the ®-power distribution, where some properties are studied, estimating approaches are developed as well as corrected bias estimator developed. We also develop censored and uncensored regression for the ®-power model and for the log-linear Birnbaum-Saunders regression models, for which model validation techniques are studied. Finally a multivariate extension of the ®-power model is proposed and some estimation procedures are investigated for the model. All the situations investigated were illustrated with data application using data sets previally analysed with other distributions.

Em analise de dados que apresentam certo grau de assimetria a suposicao que as observações seguem uma distribuição normal, pode resultar ser uma suposição irreal e a aplicação deste modelo pode ocultar características importantes do modelo verdadeiro. Este tipo de situação deu forca á aplicação de modelo assimétricos, destacando-se entre estes a família de distribuições skew-symmetric, desenvolvida por Azzalini (1985). Neste trabalho nos apresentamos uma segunda proposta para a anàlise de dados com presença importante de assimetria e/ou curtose, comparado com a distribuição normal. Nós apresentamos e estudamos algumas propriedades dos modelos alfa-potência e log-alfa-potência, onde também estudamos o problema de estimação, as matrizes de informação observada e esperada de Fisher e o grau do viés dos estimadores mediante alguns processos de simulação. Nós introduzimos um modelo mais estável que o modelo alfa- potência do qual derivamos o caso bimodal desta distribuição e introduzimos os modelos bimodal simêtrico e assimêtrico alfa-potencia. Posteriormente nós estendemos a distribuição alfa-potência para o caso do modelo Birnbaum-Saunders, estudamos as propriedades deste novo modelo, desenvolvemos estimadores para os parametros e propomos estimadores com viés corrigido. Também introduzimos o modelo de regressão alfa-potência para dados censurados e não censurados e para o modelo de regressão log-linear Birnbaum-Saunders; aqui nós derivamos os estimadores dos parâmetros e estudamos algumas técnicas de validação dos modelos. Por ultimo nós fazemos a extensão multivariada do modelo alfa-potência e estudamos alguns processos de estimação dos parâmetros. Para todos os casos estudados apresentam-se ilustrações com dados já analisados previamente com outras suposições de distribuições. / In data analysis where data present certain degree of asymmetry the assunption of normality can result in an unreal situation and the application of this model can hide important caracteristics of the true model. Situations of this type has given strength to the use of asymmetric models with special emphasis on the skew-symmetric distribution developed by Azzalini (1985). In this work we present an alternative for data analysis in the presence of signi¯cant asymmetry or kurtosis, when compared with the normal distribution, as well as other situations that involve such model. We present and study of the properties of the ®-power and log-®-power distributions, where we also study the estimation problem, the observed and expected information matrices and the degree of bias in estimation using simulation procedures. A °exible model version is proposed for the ®-power distribution, following an extension to a bimodal version. Follows next an extension of the Birnbaum-Saunders distribution using the ®-power distribution, where some properties are studied, estimating approaches are developed as well as corrected bias estimator developed. We also develop censored and uncensored regression for the ®-power model and for the log-linear Birnbaum-Saunders regression models, for which model validation techniques are studied. Finally a multivariate extension of the ®-power model is proposed and some estimation procedures are investigated for the model. All the situations investigated were illustrated with data application using data sets previally analysed with other distributions.

Evidence for a human contribution to climatic changes during the past century is accumulating rapidly. Given the strength of the evidence, it seems natural to ask whether forcing projections can be used to forecast climate change. A Bayesian method for post-processing forced climate model simulations that produces probabilistic hindcasts of inter-decadal temperature changes on large spatial scales is proposed. Hindcasts produced for the last two decades of the 20th century are shown to be skillful. The suggestion that skillful decadal forecasts can be produced on large regional scales by exploiting the response to anthropogenic forcing provides additional evidence that anthropogenic change in the composition of the atmosphere has influenced our climate. In the absence of large negative volcanic forcing on the climate system (which cannot presently be forecast), the global mean temperature for the decade 2000-2009 is predicted to lie above the 1970-1999 normal with probability 0.94. The global mean temperature anomaly for this decade relative to 1970-1999 is predicted to be 0.35C (5-95% confidence range: 0.21C-0.48C). Reconstruction of temperature variability of the past centuries using climate proxy data can also provide important information on the role of anthropogenic forcing in the observed 20th century warming. A state-space model approach that allows incorporation of additional non-temperature information, such as the estimated response to external forcing, to reconstruct historical temperature is proposed. An advantage of this approach is that it permits simultaneous reconstruction and detection analysis as well as future projection. A difficulty in using this approach is that estimation of several unknown state-space model parameters is required. To take advantage of the data structure in the reconstruction problem, the existing parameter estimation approach is modified, resulting in two new estimation approaches. The competing estimation approaches are compared based on theoretical grounds and through simulation studies. The two new estimation approaches generally perform better than the existing approach. A number of studies have attempted to reconstruct hemispheric mean temperature for the past millennium from proxy climate indicators. Different statistical methods are used in these studies and it therefore seems natural to ask which method is more reliable. An empirical comparison between the different reconstruction methods is considered using both climate model data and real-world paleoclimate proxy data. The proposed state-space model approach and the RegEM method generally perform better than their competitors when reconstructing interannual variations in Northern Hemispheric mean surface air temperature. On the other hand, a variety of methods are seen to perform well when reconstructing decadal temperature variability. The similarity in performance provides evidence that the difference between many real-world reconstructions is more likely to be due to the choice of the proxy series, or the use of difference target seasons or latitudes, than to the choice of statistical method.

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.

Το πρόβλημα που αντιμετωπίζει η διατριβή αφορά στη μοντελοποίηση μη-στασίμων τυχαίων ταλαντώσεων επί τη βάσει μετρήσεων του σήματος της ταλάντωσης, μέσω μοντέλων FS-TAR/TARMA. Οι στόχοι της διατριβής περιλαμβάνουν την αποτίμηση της εφαρμοσιμότητας των μεθόδων FS-TAR/TARMA για την μοντελοποίηση και ανάλυση της ταλάντωσης χρονικά μεταβαλλόμενών κατασκευών, καθώς και τη σύγκρισή τους με εναλλακτικές παραμετρικές μεθόδους του πεδίου του χρόνου. Ιδιαίτερη βαρύτητα δίνεται και στην αντιμετώπιση θεμάτων που σχετίζονται με την εκτίμηση μοντέλων FS-ΤAR/TARMA, καθώς και στην θεωρητική ασυμπτωτική ανάλυση των ιδιοτήτων των εκτιμητριών που χρησιμοποιούνται. Η διατριβή αρχικά παρουσιάζει μια συγκριτική ανασκόπηση της βιβλιογραφίας στο θέμα της μοντελοποίησης μη-στασίμων ταλαντώσεων μέσω παραμετρικών μεθόδων του πεδίου του χρόνου, η οποία και επιδεικνύει τα πλεονεκτήματα των μεθόδων FS-TAR/TARMA. Στη συνέχεια αντιμετωπίζεται μια σειρά προβλημάτων που εμφανίζονται κατά την εκτίμηση (των παραμέτρων) και την επιλογή της δομής του μοντέλου. Η αποτελεσματικότητα των μεθόδων FS-TAR/TARMA για την μοντελοποίηση και ανάλυση μη-στάσίμων ταλαντώσεων επιδεικνύεται και πειραματικά μέσω εφαρμογής στην οποία πραγματοποιείται επιτυχής εξαγωγή των δυναμικών χαρακτηριστικών μιας εργαστηριακής χρονικά μεταβαλλόμενης κατασκευής. Στη συνέχεια, η διατριβή εστιάζει στην αναζήτηση ακριβέστερων εκτιμητριών, καθώς και στην ασυμπτωτική ανάλυση των ιδιοτήτων των εκτιμητριών «γενικών» (όχι αναγκαστικά περιοδικά μεταβαλλόμενων) μοντέλων FS-TAR/TARMA. Συγκεκριμένα, εξετάζονται οι περιπτώσεις των εκτιμητριών σταθμισμένων ελαχίστων τετραγώνων [Weighted Least Squares (WLS)], μέγιστης πιθανοφάνειας [Maximum Likelihood (ML)], καθώς και μια εκτιμήτρια πολλαπλών σταδίων [Multi Stage (MS)], η οποία αναπτύσσεται στην παρούσα διατριβή και είναι ασυμπτωτικά ισοδύναμη με την εκτιμήτρια ML ενώ ταυτόχρονα χαρακτηρίζεται από μειωμένη υπολογιστική πολυπλοκότητα. Στη διατριβή αποδεικνύεται η συνέπεια (consistency) των εκτιμητριών αυτών και εξάγεται η ασυμπτωτική κατανομή (asymptotic distribution) τους. Παράλληλα, αναπτύσσεται μια συνεπής εκτιμήτρια του ασυμπτωτικού πίνακα συνδιασποράς και μια μέθοδος για τον έλεγχο εγκυρότητας των μοντέλων FS-TAR/TARMA. Η ορθότητα των αποτελεσμάτων της ασυμπτωτικής ανάλυσης επιβεβαιώνεται μέσω μελετών Monte Carlo. / The thesis studies the problem of non-stationary random vibration modeling and analysis based on available measurements of the vibration signal via Functional Series Time-dependent AutoRegressive / AutoRegressive Moving Average (FS-TAR/ TARMA) models. The aims of the thesis include the assessment of the applicability of FS-TAR/TARMA methods for the modeling and analysis of non-stationary random vibration, as well as their comparison with alternative time-domain parametric methods. In addition, significant attention has been paid to the FS-TAR/TARMA estimation problem and to the theoretical asymptotic analysis of the estimators. A critical overview and comparison of time-domain, parametric, non-stationary random vibration modeling and analysis methods is firstly presented, where the high potential of FS-TAR/TARMA methods is demonstrated. In the following, a number of issues concerning the FS-TAR/TARMA model (parameter) estimation and model structure selection are considered. The effectiveness of the FS-TARMA methods for non-stationary random vibration modeling and analysis is experimentally demonstrated, through their application for the recovery of the dynamical characteristics of a time-varying bridge-like laboratory structure. In the sequel, the thesis focuses on the asymptotic analysis of “general” (that is not necessarily periodically evolving) FS-TAR/TARMA estimators. In particular, the Weighted Least Squares (WLS) and Maximum Likelihood (ML) estimators are both investigated, while a Multi Stage (MS) estimator, that approximates the ML estimator at reduced complexity, is developed. The consistency of the considered estimators is established and their asymptotic distribution is extracted. Furthermore, a consistent estimator of the asymptotic covariance matrix is formulated and an FS-TAR/TARMA model validation method is proposed. The validity of the theoretical asymptotic analysis results is assessed through several Monte Carlo studies.

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Testing Structural Changes Using Ratio Type Statistics Barbora Peštová Charles University in Prague, Faculty of Mathematics and Physics, Department of Probability and Mathematical Statistics, Czech Republic Abstract of the doctoral thesis We deal with sequences of observations that are naturally ordered in time and assume various underlying stochastic models. These models are parametric and some of the parameters are possibly subject to change at some unknown time point. The main goal of this thesis is to test whether such an unknown change has occurred or not. The core of the change point methods presented here is in ratio type statistics based on maxima of cumulative sums. Firstly, an overview of thesis' starting points is given. Then we focus on methods for detecting a gradual change in mean. Consequently, procedures for detection of an abrupt change in mean are generalized by considering a score function. We explore the possibility of applying the bootstrap methods for obtaining critical values, while disturbances of the change point model are considered as weakly dependent. Procedures for detection of changes in parameters of linear regression models are shown as well and a permutation version of the test is derived. Then, a related problem of testing a change in autoregression parameter is studied....

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