Repni indeks i zavisnost

Cilj ovog rada bio je istražiti metode procjene repnog indeksa te mjera zavisnosti kod distribucija teškog repa te ih isprobati u praksi, ali i upozoriti na probleme u njihovoj primjeni. Uveli smo pojam regularne varijacije iz matematičke analize i slabe i slabašne konvergencije iz teorije mjere te smo iskazali neka njihova svojstva. Zatim smo definirali Hillov procjenitelj za repni indeks \(\alpha\) slučajne varijable čija je funkcija doživljenja regularno varirajuća i dokazali njegovu konzistentnost. Da bismo ga isprobali u praksi, koristili smo slučajno generirane podatke iz Paretove, normalne i Cauchyjeve distribucije te podatke o dnevnom dolarskom volumenu dionica nekolicine tehnoloških kompanija. Tu su se pojavili slični problemi na koje literatura upozorava - nije lagano uopće ustvrditi kada je model teškog repa primjeren, a čak niti jednom kad smo ga odlučili koristiti nije jasno kako točno interpretirati Hillov procjenitelj, odnosno koji prag koristiti. Za neke je dionice Hillov graf jasno sugerirao vrijednost \(\alpha\) između 3 i 3:5, dok za neke to nije bilo jednostavno iščitati. Kako bismo mogli govoriti o repnoj zavisnosti slučajnih vektora, morali smo definirati višedimenzionalnu generalizaciju regularne varijacije i iskazati neka njezina svojstva. Zatim smo definirali tri mjere repne zavisnosti - kutnu mjeru (s kojom je teško raditi na konačnom uzorku jer se procjenjuje mjera na jediničnoj sferi, beskonačnodimenzionalan objekt), te koeficijente \(\chi\) i \(\bar{\chi}\) koji imaju slične probleme izbora praga kao Hillov procjenitelj. Njihove su procjene prilično dobro funkcionirale na generiranim podacima, te su dali procjenu vrijednosti \(\chi\) blizu nule za svaki par dionica, a \(\bar{\chi}\) za različite parove između 0 i 0:3.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