## Συναρτήσεις copula (Master thesis)

#### Ταγκαρέλη, Ευαγγελία

The concepts of dependence or independence between two or more random variables is of fundamental importance in statistics and probability theory since independence in the data is always a desirable property (cf. Micheas and Zografos, 2006). In this context, copula theory provides us with tools to formulate dependence or independence in the data. Therefore, the main aim of this thesis is to present and discuss the most basic results of copula theory by following the existing literature, as it is mentioned in the specific places in the text. Copulas is a relatively recent subject in probability theory and statistics and they have been developed to formulate correlated multivariate data, arising in several disciplines and contexts. Following Fisher (1997), in his article in the Encyclopedia of Statistical Sciences, "Copulas are of interest to statisticians for two main reasons: Firstly, as a way of studying scale-free measures of dependence, and secondly, as a starting point for constructing families of bivariate distributions, ...". The notion of copula was introduced by Abe Sklar in 1959 and the main theorem which inspired the definition of copula function has the name of Sklar. Sklar used the term copula since this function provides a link between the marginal distributions and the joint distribution of two or more random variables. Albeit Sklar formulated the definition of copula, this does not necessarily mean that the copula functions did not exist before. The idea of the copula function has been previously appeared in the papers by Hoeffding (1940, 1941), which were mainly focused on two-dimensional standard distributions. The aforementioned work has moreover established the best possible limits for copula functions which were also elaborated in the papers by Fréchet (1951, 1958). Many authors, such as, Gumbel, Plackett, Mardia, Ali, Mikhail and Haq, Clayton, Joe, Genest, Nelsen, dealt with copulas and some of their contributions in the field appeared in the references. The aim of this thesis is a presentation of the most important results which are related to copulas. More specifically, Chapter 2 is concentrated to Sklar’s theorem, which is of fundamental importance in copula theory. It is also concentrated to the definition of the copula function and to the Fréchet- Hoeffding bounds. Some basic properties of the copulas are presented and reference is made to the notion of copula density and to the notion of survival copulas. Some known families of copulas are discussed, in the sequel, and the interest is, moreover, focused in the definition and study of some correlation measures, namely the Pearson linear correlation coefficient, the Spearman’s Rho and Kendall’s Tau correlation coefficients. Chapter 3 deals with the two-dimensional extreme value copula. At the beginning, the definition of this specific family of copulas is presented and, then, the connection of this copula with the concept of max-stable copulas is discussed. Then, the Pickand’s (1981) process of constructing copulas of extreme values, is given. In addition, the interest is focused on some special cases of Pickands Dependence and this chapter is finished by a tabulation of the Pickands dependence function of some known copula families and the respective Kendall and Spearman correlation coefficients. The most general class of copula functions, that is the Archimedean copulas, are introduced in Chapter 4. Initially, some basic definitions, properties and results related to Archimedean copulas are presented. Then, the level sets of Archimedean copulas and the respective densities, in the bivariate case, are formulated. Dependency and correlation measures are presented and Clayton, Frank and Gumbel families of copulas are discussed.
 Alternative title / Subtitle: μια ανασκόπηση Institution and School/Department of submitter: Πανεπιστήμιο Ιωάννινων. Σχολή Θετικών Επιστημών. Τμήμα Μαθηματικών Subject classification: Copulas (Mathematical statistics) Keywords: Copulas επιβίωσης,Οικογένειες συναρτήσεων copulas,Copulas ακραίων τιμών,Εξάρτηση και συσχέτιση και συνάρτηση copula,Copulas,Survival copulas,Archimedean copulas,Frechet - Hoeffding URI: https://olympias.lib.uoi.gr/jspui/handle/123456789/29297http://dx.doi.org/10.26268/heal.uoi.2953 Appears in Collections: Διατριβές Μεταπτυχιακής Έρευνας (Masters)

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