# Repni indeks i zavisnost

Stipetić, Ognjen (2016) Repni indeks i zavisnost. Diploma thesis, Faculty of Science > Department of Mathematics.

 Preview
PDF
Language: Croatian

Archive (dodatni materijali ("kodovi"))
Language: Croatian

The aim of this thesis was to study estimators of the tail index and measures of dependence for heavy-tailed random variables and to test those methods on some real life data, but also to discuss some problems in this context. We introduced the concept of regular variation from mathematical analysis as well as weak and vague convergence from measure theory and stated some of their properties. We defined Hill's estimator for the tail index of a random variable whose tail distribution is regularly varying and proved its consistency. We used randomly generated data from Pareto, normal and Cauchy distributions to illustrate performance of Hill's estimator and then applied it to tech company stocks daily dollar volume data to see how it works in practice. We encountered the same problems literature warns us about - it may not be simple to determine whether such a heavy tail model is appropriate, and even once we decided to use it, it's not easy to select appropriate threshold in the corresponding Hill plot. For some stocks Hill plot clearly suggests value of $\alpha$ in the interval between 3 and 3:5, for others it is very hard conclude anything. We introduced multivariate regular variation in order to measure tail dependence between components of random vectors. We defined three measures of tail dependence – angular measure (which is hard to use on a finite sample because, as a measure on unit sphere, it is very difficult to estimate), and coefficients $\chi$ and $\bar{\chi}$ which have problems with choosing threshold similar to the Hill estimator. Standard estimators of the quantities worked quite well on simulated data from bivariate logistic extreme value distribution, and when applied to the stocks, gave us estimates for $\chi$ very close to 0 for each pair of stocks, and $\bar{\chi}$ varying between 0 and 0:3.