A Tail Plot Without Moment Restrictions

Abstract

This thesis studies heavy-tailed probability distributions with a focus on the Pareto tail plot. The main aim is to investigate an approach that enables the identification of Pareto-type tail be- haviour without requiring finite moments, in contrast to the mean excess plot. The theoretical part introduces the main concepts, presents the Pareto tail plot, and includes a detailed proof of a key characterisation theorem. Furthermore, an estimator based on U-statistics is discussed, and its asymptotic distribution is derived. The practical part examines the method through a simu- lation study and an application to real data. The results demonstrate that the proposed approach provides a more reliable tool for tail analysis, particularly in settings with extremely heavy-tailed distributions.

Tato bakala ́r ̌ska ́ pra ́ce se zaby ́va ́ analy ́zou pravde ̌podobnostnı ́ch rozde ̌lenı ́ s te ̌z ̌ky ́mi chvosty se zame ̌r ̌enı ́m na Pareto tail plot. Hlavnı ́m cı ́lem je zkoumat pr ̌ı 'stup, ktery ́ umoz ̌n ̌uje identifikaci chova ́nı ́ chvostu Paretova typu bez nutnosti existence konec ̌ny ́ch momentu ̊, na rozdı ́l od grafu str ̌ednı ́ho pr ̌esahu. Teoreticka ́ c ̌a 'st pr ̌edstavuje za ́kladnı ́ pojmy, popisuje Pareto tail plot a obsahuje detailnı ́ du ̊kaz klı ́c ̌ove ́ charakterizac ̌nı ́ ve ̌ty. Da ́le je diskutova ́n odhad zaloz ̌eny ́ na U-statistika ́ch a je uvedeno odvozenı ́ jeho asymptoticke ́ho rozde ̌lenı ́. Prakticka ́ c ̌a 'st zkouma ́ vlastnosti metody pomocı ́ simulac ̌nı ́ studie a aplikace na rea ́lna ́ data. Vy 'sledky ukazujı ́, z ̌e navrz ̌eny ́ pr ̌ı 'stup poskytuje spolehlive ̌js ̌ı ́ na 'stroj pro analy ́zu chvostu ̊, zejme ́na v situacı ́ch s extre ́mne ̌ te ̌z ̌ky ́mi chvosty.