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Adaptive designs in phase II clinical trials
(23-09-2013) Poulopoulou, Stavroula; Πουλοπούλου, Σταυρούλα; Athens University of Economics and Business. Department of Statistics; Karlis, Dimitrios; Dafni, Urania
Clinical trials play a very important role in the development process of new therapies. Recently there has been a rapid increase in theresearch and creation of new modern molecular agents, which makes necessary the development of more flexible and adaptive designs forthe implementation of clinical trials. The objective of adaptive designs is to ensure direct and dynamic control of the effectiveness and thesafety of a new treatment by allowing the adjustment of the elements of the study (i.e sample size), during the study, in such a way that wewill not sacrifice elements which are associated with the credibility of the study (i.e statistical power) and also issues which concern ethicalcharacteristics of the clinical trials.
Application of Copula functions in statistics
(09-2007) Nikoloulopoulos, Aristidis; Νικολουλόπουλος, Αριστείδης; Athens University of Economics and Business, Department of Statistics; Karlis, Dimitrios
Studying associations among multivariate outcomes is an interesting problem in statistical science. The dependence between random variables is completely described by their multivariate distribution. When the multivariate distribution has a simple form, standard methods can be used to make inference. On the other hand one may create multivariate distributions based on particular assumptions, limiting thus their use. Unfortunately, these limitations occur very often when working with multivariate discrete distributions. Some multivariate discrete distributions used in practice can have only certain properties, as for example they allow only for positive dependence or they can have marginal distributions of a given form. To solve this problem copulas seem to be a promising solution. Copulas are a currently fashionable way to model multivariate data as they account for the dependence structure and provide a flexible representation of the multivariate distribution. Furthermore, for copulas the dependence properties can be separated from their marginal properties and multivariate models with marginal densities of arbitrary form can be constructed, allowing a wide range of possible association structures. In fact they allow for flexible dependence modelling, different from assuming simple linear correlation structures. However, in the application of copulas to discrete data marginal parameters affect dependence structure, too, and, hence the dependence properties are not fully separated from the marginal properties. Introducing covariates to describe the dependence by modelling the copula parameters is of special interest in this thesis. Thus, covariate information can describe the dependence either indirectly through the marginal parameters or directly through the parameters of the copula . We examine the case when the covariates are used both in marginal and/or copula parameters aiming at creating a highly flexible model producing very elegant dependence structures. Furthermore, the literature contains many theoretical results and families of copulas with several properties but there are few papers that compare the copula families and discuss model selection issues among candidate copula models rendering the question of which copulas are appropriate and whether we are able, from real data, to select the true copula that generated the data, among a series of candidates with, perhaps, very similar dependence properties. We examined a large set of candidate copula families taking intoaccount properties like concordance and tail dependence. The comparison is made theoretically using Kullback-Leibler distances between them. We have selected this distance because it has a nice relationship with log-likelihood and thus it can provide interesting insight on the likelihood based procedures used in practice. Furthermore a goodness of fit test based on Mahalanobis distance, which is computed through parametric bootstrap, will be provided. Moreover we adopt a model averaging approach on copula modelling, based on the non-parametric bootstrap. Our intention is not to underestimate variability but add some additional variability induced by model selection making the precision of the estimate unconditional on the selected model. Moreover our estimates are synthesize from several different candidate copula models and thus they can have a flexible dependence structure. Taking under consideration the extended literature of copula for multivariate continuous data we concentrated our interest on fitting copulas on multivariate discrete data. The applications of multivariate copula models for discrete data are limited. Usually we have to trade off between models with limited dependence (e.g. only positive association) and models with flexible dependence but computational intractabilities. For example, the elliptical copulas provide a wide range of flexible dependence, but do not have closed form cumulative distribution functions. Thus one needs to evaluate the multivariate copula and, hence, a multivariate integral repeatedly for a large number of times. This can be time consuming but also, because of the numerical approach used to evaluate a multivariate integral, it may produce roundoff errors. On the other hand, multivariate Archimedean copulas, partially-symmetric m-variate copulas with m-1 dependence parameters and copulas that are mixtures of max-infinitely divisible bivariate copulas have closed form cumulative distribution functions and thus computations are easy, but allow only positive dependence among the random variables.The bridge of the two above-mentioned problems might be the definition of a copula family which has simple form for its distribution functionwhile allowing for negative dependence among the variables. We define sucha multivariate copula family exploiting the use of finite mixture of simple uncorrelated normal distributions. Since the correlation vanishes, the cumulative distribution is simply the product of univariate normal cumulative distribution functions. The mixing operation introduces dependence. Hence we obtain a kind of flexible dependence, and allow for negative dependence.
Application of hidden Markov and related models to earthquake studies
(2015) Orfanogiannaki, Aikaterini M.; Ορφανογιαννάκη, Αικατερίνη Μ.; Athens University of Economics and Business, Department of Statistics; Karlis, Dimitrios
Discrete valued hidden Markov Models (HMMs) are used to model time series of event counts in several scientific fields like genetics, engineering, seismology and finance. In its general form the model consists of two parts: the observation sequence and an unobserved sequence of hidden states that underlies the data and consist a Markov chain. Each state is characterized by a specific distribution and the progress of the hidden process from state to state is controlled by a transition probability matrix. We extend the theory of HMMs to the multivariate case and apply them to seismological data fromdifferent seismotectonic environments. This extension is not straightforward and it is achieved gradually by assuming different multivariate distributions to describe each state of the model.
Model based clustering for count and mixed mode data
(28-09-2022) Πανάγου, Φωτεινή; Panagou, Fotini; Athens University of Economics and Business, Department of Statistics; Papageorgiou, Ioulia; Gormley, Claire; Kosmidis, Ioannis; Rau, Andrea; Ntzoufras, Ioannis; Papastamoulis, Panagiotis; Karlis, Dimitrios
Οι μέθοδοι ομαδοποίησης που βασίζονται σε μοντέλα κατανομής για τον πληθυσμό, είναι μια κοινή προσέγγιση για τη μοντελοποίηση δεδομένων με τη χρήση πεπερασμένων μίξεων παραμετρικών κατανομών. Για μετρήσιμα δεδομένα, η επιλογή της πολυμεταβλητής κατανομής Poisson μπορεί να οδηγήσει σε αυξημένο υπολογιστικό κόστος. Η έννοια της μεθόδου της σύνθετης πιθανοφάνειας με τη χρήση διμεταβλητών περιθώριων κατανομών μπορεί να προσφέρει ευελιξία στις εκτιμήσεις. Προκειμένου να μειωθεί περαιτέρω ο χρόνος εκτίμησης των παραμέτρων που σχετίζονται με τη σύνθετη μέθοδο πιθανοφάνειας, εισάγουμε μεθόδους δειγματοληψίας που μπορούν να προσφέρουν επαρκή αποτελέσματα, ειδικά σε μεγάλων διαστάσεων δεδομένα. Όσον αφορά τα δεδομένα μεικτού τύπου, η από κοινού κατανομή δεν είναι πάντα εύκολο να βρεθεί. Τα copulas είναι ευρέως γνωστά ως ευέλικτα μοντέλα που επιτρέπουν τη δημιουργία πολυμεταβλητών κατανομών όταν δίνονται οι περιθώριες κατανομές. Ως εκ τούτου, μπορούν να δημιουργήσουν μια πληθώρα πολυμεταβλητών μοντέλων συμπεριλαμβανομένων μοντέλων με διαφορετικές περιθώριες. Σκοπός της παρούσας διπλωματικής εργασίας είναι κυρίως να επεκτείνει τα μέχρι τώρα αποτελέσματα της χρήσης μοντέλων που βασίζονται σε copula για εφαρμογές ομαδοποίησης. Το Gaussian Copula προσφέρει ευελιξία για την περιγραφή των συσχετίσεων μεταξύ διαφορετικών τύπων μεταβλητών. Στόχος μας είναι να μειώσουμε περαιτέρω το υπολογιστικό κόστος που προκύπτει από τη χρήση του Gaussian copula και του πλήρως παραμετροποιημένου μοντέλου που μελετήσαμε εκτενώς, καθώς αυτή η προσέγγιση είναι χρονοβόρα, γεγονός που προκύπτει από την προσθήκη διαφορετικών πινάκων συσχέτισης για κάθε ομάδα που πρέπει να εκτιμηθεί. Έτσι, ο κύριος στόχος είναι να επιτευχθεί ευελιξία στην εκτίμηση με τη χρήση κατάλληλων τεχνικών. Στην παρούσα διατριβή έχουμε προτείνει ευέλικτες εναλλακτικές που βασίζονται σε προσεγγίσεις μείωσης των διαστάσεων, όπως η ανάλυση παραγόντων ή έξυπνες αναπαραστάσεις των πινάκων συσχέτισης (δομημένοι πίνακες συσχέτισης).
Modeling multivariate time series
(21-11-2023) Tsamtsakiri, Panagiota; Τσαμτσακίρη, Παναγιώτα; Athens University of Economics and Business, Department of Statistics; Ntzoufras, Ioannis; Vrontos, Ioannis; Pedeli, Xanthi; Fokianos, Konstantinos; Ombao, Hernando; Barretto-Souza, Wanger; Karlis, Dimitrios
This thesis deals with the construction of multivariate Integer Generalized Au toregressive Conditional Heteroskedastic(INGARCH) and Conditional Autoregres sive Range(CARR) time series processes. The INGARCH(1,1) model has been alsostudied in multivariate case and implementations in 2 dimensions have been also pre sented recent years. The drawback was found at the restricted values of correlationcoefficient boundaries depending on the way of model’s construction. Based on afamily of copulas a multivariate INGARCH(1,1) model and a CARR(1,1) model arestudied considering interdependencies and self-dependencies respectively. Accordingto model’s complexity, its appropriateness and capability to study data with smallsample sizes are examined and provided with simulations. H-steps ahead forecast ing is considered by taking conditional expectation on volatilities and calculatingmarginal probability mass function.Firstly, considering univariate INGARCH models where volatilities are linearly orlog-linearly expressed offering more flexibility on conditions of stationarity respec tively, a Bayesian Trans-dimensional Markov Chain Monte Carlo is provided. At asecond stage a new alternative family of Sarmanov distribution is also presented inorder to ameliorate boundaries of correlation coefficient and comparison with theknown Sarmanov families are graphically discussed. A multivariate INGARCH(1,1)process is studied based on this alternative Sarmanov distribution and an imple mentation with daily crash counts on three towns in Netherlands is presented. Amultivariate Conditional Aytoregressive Range(CARR(1,1)) model assuming an ex ponential distribution and reconstructing correlation coefficients boundaries is dis cussed. The proposed model is illustrated on bivariate series from the fields ofrenewable sources.

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