Bayesian Inference for Cure Rate Models

Η ανάλυση επιβίωσης αποτελείται από ένα σύνολο στατιστικών μεθόδων που στοχεύει στη μελέτη του χρόνου μέχρι την εμφάνιση ενός συγκεκριμένου γεγονότος όπως ο θάνατος. Για την πλειονότητα των μεθόδων αυτών, θεωρείται πως όλα τα άτομα που συμμετέχουν υπόκεινται στο γεγονός που μας ενδιαφέρει. Ωστόσο, υπάρχουν περιπτώσεις όπου η υπόθεση αυτή δεν είναι ρεαλιστική, καθώς υπάρχουν ασθενείς που δεν θα βιώσουν το γεγονός αυτό στη διάρκεια της μελέτης. Για αυτό το λόγο, έχουν αναπτυχθεί ορισμένα μοντέλα επιβίωσης που επιτρέπουν την ύπαρξη ασθενών οι οποίοι δε βιώνουν το συμβάν και ονομάζονται μακροχρόνια επιζώντες. Τα μοντέλα αυτά ονομάζονται μοντέλα ρυθμού θεραπείας και υποθέτουν ότι, καθώς ο χρόνος αυξάνεται, η συνάρτηση επιβίωσης τείνει σε μια τιμή p ∈ (0,1), που αντιπροσωπεύει το ποσοστό των μακροχρόνια επιζώντων, αντί να τείνει στο μηδέν όπως στην κλασική ανάλυση επιβίωσης. Πρόσφατα, ο Rocha (2016) πρότεινεμία νέα προσέγγισητωνπροβλημάτωνεπιβίωσης μεμακροχρόνια επιζώντες. Η μεθοδολογία του για τη μοντελοποίηση του ποσοστού των μακροχρόνια επιζώντων βασίστηκε στη χρήση των «ελαττωματικών» (defective) κατανομών, οι οποίες χαρακτηρίζονται από το γεγονός ότι το ολοκλήρωμα της συνάρτησης πιθανότητάς τους δεν ισούται με τη μονάδα για ορισμένες επιλογές του πεδίου ορισμού κάποιων παραμέτρων τους. Σκοπός της παρούσας διπλωματικής εργασίας, είναι να παράσχει νέους Μπεϋζιανούς εκτιμητές των παραμέτρων των «ελαττωματικών» μοντέλων κάτω από την υπόθέση της δεξιάς λογοκρισίας. Επίσης, θα αναπτυχθούν αλγόριθμοι Markov chain Monte Carlo (MCMC) για τη συμπερασματολογία σχετικά με τις παραμέτρους μιας ευρείας κατηγορίας μοντέλων ρυθμού θεραπείας βασισμένων στις «ελαττωματικές» αυτές κατανομές, ενώ οι Μπεϋζιανοί εκτιμητές και τα αντίστοιχα διαστήματα αξιοπιστίας θα ληφθούν από τα δείγματα της από κοινού εκ των υστέρων κατανομής. Επιπλέον, η συμπεριφορά των Μπεϋζιανών εκτιμητών θα αξιολογηθεί και θα συγκριθεί με αυτή των εκτιμητών μεγίστης πιθανοφάνειας του Rocha (2016) μέσω πειραμάτων προσομοίωσης. Ακόμη, τα προτεινόμενα αυτά μοντέλα-κατανομές θα εφαρμοσθούν σε πραγματικά σετ δεδομένων, όπου και θα συγκριθούν μεταξύ τους μέσω κατάλληλων στατιστικών μεγεθών. Τέλος, αξίζει να σημειωθεί πως η παρούσα διπλωματική εργασία αποτελεί την πρώτη προσπάθεια διερεύνησης των πλεονεκτημάτων της Μπεϋζιανής προσέγγισης στη συμπερασματολογία για τις παραμέτρους αρκετών μοντέλων ρυθμού θεραπείας, κάτω από την υπόθεση της δεξιάς λογοκρισίας, καθώς και της απόκτησης νέων Μπεϋζιανών εκτιμητών, χωρίς όμως τη συμπερίληψη της πληροφορίας από συν μεταβλητές.Survival analysis consists of a set of statistical methods in the field of biostatistics, whose main aim is to study the time until the occurrence of a specified event, such as death. For the majority of these methods it is assumed that all the individuals taking part in the study are subject to the event of interest. However, there are situations where this assumption is unrealistic, since there are observations not susceptible to the event of interest or cured. For this reason, there have been developed some survival models which allow for patients that may never experience the event, usually called long-term survivors. These models, called Cure Rate Models, assume that, as time increases, the survival function tends to a value p ∈ (0,1), representing the cure rate, instead of tending to zero as in standard survival analysis. Recently, Rocha (2016) proposed a new approach to modelling the situations in which there are long-term survivors in survival studies. His methodology was based on the use of defective distributions to model cure rates. In contrast to the standard distributions, the defective ones are characterized by having probability density functions which integrate to values less than one for certain choices of the domain of some of their parameters. The aim of the present thesis is to provide new Bayesian estimates for the parameters of the defective models used for cure rate modelling under the assumption of right censoring. We will develop Markov chain Monte Carlo (MCMC) algorithms for inferring the parameters of a broad class of defective models, both for the baseline distributions (Gompertz & Inverse Gaussian), as well as, for their extension under the Marshall-Olkin family of distributions. The Bayesian estimates of the distributions’ parameters, as well as their associated credible intervals, will be obtained from the samples drawn from their joint posterior distribution. In addition, Bayesian estimates’ behaviour will be evaluated and compared with the maximum likelihood estimates obtained by Rocha (2016) through simulation experiments. Finally, we will apply the competing models and approaches to real datasets and we will compare them through various statistical measures. This work will be the first attempt to explore the advantages of the Bayesian approach to inference for defective cure rate models under the assumption of right censoring mechanism, as well as the first presentation of new Bayesian estimates for several defective distributions, but without incorporating covariate information

Characterizations and Infinite Divisibility of Certain Recently Introduced Distributions IV

Certain characterizations of recently proposed univariate continuous distributions are presented in different directions. This work contains a good number of reintroduced distributions and may serve as a source of preventing the reinvention and/or duplication of the existing distributions in the future

The Burr X Exponentiated Weibull Model: Characterizations, Mathematical Properties and Applications to Failure and Survival Times Data

In this article, we introduce a new three-parameter lifetime model called the Burr X exponentiated Weibull model. The major justification for the practicality of the new lifetime model is based on the wider use of the exponentiated Weibull and Weibull models. We are motivated to propose this new lifetime model because it exhibits increasing, decreasing, bathtub, J shaped and constant hazard rates. The new lifetime model can be viewed as a mixture of the exponentiated Weibull distribution. It can also be viewed as a suitable model for fitting the right skewed, symmetric, left skewed and unimodal data. We provide a comprehensive account of some of its statistical properties. Some useful characterization results are presented. The maximum likelihood method is used to estimate the model parameters. We prove empirically the importance and flexibility of the new model in modeling two types of lifetime data. The proposed model is a better fit than the Poisson Topp Leone-Weibull, the Marshall Olkin extended-Weibull, gamma-Weibull , Kumaraswamy-Weibull , Weibull-Fréchet, beta-Weibull, transmuted modified-Weibull, Kumaraswamy transmuted- Weibull, modified beta-Weibull, Mcdonald-Weibull and transmuted exponentiated generalized-Weibull models so it is a good alternative to these models in modeling aircraft windshield data as well as the new lifetime model is much better than the Weibull-Weibull, odd Weibull- Weibull, Weibull Log-Weibull, the gamma exponentiated-exponential and exponential exponential-geometric models so it is a good alternative to these models in modeling the survival times of Guinea pigs. We hope that the new distribution will attract wider applications in reliability, engineering and other areas of research

Bell-Touchard-G Family of Distributions: Applications to Quality Control and Actuarial Data

In this article, we developed a new statistical model named as the generalized complementary exponentiated Bell-Touchard model. The exponential model is taken as a special baseline model with a configurable failure rate function. The proposed model is based on several features of zero-truncated Bell numbers and Touchard polynomials that can address the complementary risk matters. The linear representation of the density of the proposed model is provided that can be used to obtain numerous important properties of the special model. The well-known actuarial metrics namely value at risk and expected shortfall are formulated, computed and graphically illustrated for the sub model. The maximum likelihood approach is used to estimate the parameters. Furthermore, we designed the group acceptance sampling plan for the proposed model by using the median as a quality parameter for truncated life tests. Three real data applications are offered for the complementary exponentiated Bell Touchard exponential model. The analysis of the two failure times data and comparative study yielded optimized results of the group acceptance sampling plan under the proposed model. The application to insurance claim data also provided the best results and showed that the proposed model had heavier tail

Weibull, κ-Weibull and Other Probability Distributions

Here we will consider a function of κ-statistics, the κ-Weibull distribution, and compare it to the well-known Weibull distribution. The κ-Weibull will be also compared to the 3-parameter extended Weibull function, obtained according to the Marshall–Olkin extended distributions. The log-logistic distribution will be considered for comparison too, such as the exponentiated Weibull, the Burr and the q-Weibull distributions. The most important observation, coming from the proposed calculations, is that the κ-Weibull hazard function is strongly depending on the values of parameter κ, a parameter which is deeply influencing the behaviour of the tail of the probability distribution. As a consequence, the κ-Weibull function turns out to be quite relevant for generalizations of the Weibull approach to modeling failure times. Discussions about the Maximum Likelihood approach for Weibull, κ-Weibull and Burr distributions will be also given

The Extended Burr XII Distribution with Variable Shapes for the Hazard Rate

<p>We define and study a new continuous distribution called the exponentiated Weibull Burr XII. Its density function can be expressed as a linear mixture of Burr XII. Its hazard rate is very flexibile in accomodating various shapes including constant, decreasing, increasing, J-shape, unimodal or bathtub shapes. Various of its structural properties are investigated including explicit expressions for the ordinary and incomplete moments, generating function, mean residual life, mean inactivity time and order statistics. We adopted the maximum likelihood method for estimating the model parameters. The flexibility of the new family is illustrated by means of a real data application.</p

Modified weibull distributions in reliability engineering

Ph.DDOCTOR OF PHILOSOPH