Brownov most i Kolmogorov-Smirnovljeva statistika

master's thesis

• Author: Nikolina Blažević
• Mentor: Ante Mimica (mentor)
• Mentor: Siniša Slijepčević (mentor)
• Committee member: Siniša Slijepčević (predsjednik povjerenstva)
• Committee member: Vjeran Hari (član povjerenstva)
• Committee member: Franka Miriam Bruckler (član povjerenstva)
• Committee member: Marko Vrdoljak (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2016-02-24, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovome radu uveden je pojam Brownovog mosta pomoću kojeg je izvedena asimptotska distribucija Kolmogorov-Smirnovljeve statistike koja služi za testiranje činjenice dolazi li slučajni uzorak iz neke populacije s neprekidnom razdiobom. Prvo poglavlje je kratko ponavljanje pojmova iz teorije vjerojatnosti bez kojih je teško razumjeti ostatak rada. Uvodimo pojam vjerojatnosnog prostora, slučajne varijable i nekih njenih svojstava te primjere poznatih distribucija koji će nam biti od koristi u nastavku. Definiranjem empirijske funkcije distribucije i raspisa njenih svojstava dolazimo do motivacije za nastavak rada. U drugom poglavlju definiramo Brownovo gibanje, jedan od najznačajnih stohastičkih procesa i dokazujemo teoreme koji će poistovjetiti Brownovo gibanje s Gaussovskim procesom, pokazati da zadovoljava jako Markovljevo svojstvo i kao posljedicu tog svojstva dokazujemo princip refleksije. Slijedi dokaz Lévyevog zakona trojke i iskaz principa invarijantnosti, poznatijeg kao Donskerov teorem. Na koncu je definiran Brownov most, prikazana su njegova svojstva i dokazan rezultat koji će nam biti ključan u zadnjem poglavlju, a koji nam govori da je Brownov most aproksimativno jednak Brownovom gibanju privezanom da bude u okolini nule u vremenu 1. Na kraju poglavlja imamo vezu Brownovog mosta i empirijske funkcije distribucije. Posljednje poglavlje ovog rada uvodi pojmove vezane uz testiranje statističkih hipoteza, zatim govori o Kolmogorov-Smirnovljevoj statistici definiranoj preko empirijske funkcije distribucije i prikazuje statistički test u kojem se koristi kao testna statistika. Konačno dolazimo do krajnjeg rezultata koji nam govori da Kolmogorov-Smirnovljeva statistika ima asimptotsku distribuciju i da upravo Brownov most, pa posljedično i Brownovo gibanje nam daju način kako da ju izračunamo.

Abstract (english)

This thesis introduces the concept of Brownian bridge with which is derived the asymptotic distribution of Kolmogorov-Smirnov statistic used for testing whether a random sample comes from a particular population with countinuous distribution. The first chapter is a brief introduction to the concepts of probability theory without which is difficult to understand the subject matter of this paper. Probability space, random variables along with some of its properties and examples of well-known distributions that will be useful later on, are specified in this chapter. Defining the empirical distribution function and its properties brings us the motivation for the main task of this paper. Further, the second chapter defines one of the most important stochastic processes, Brownian motion. Theorems to identify Brownian motion with Gaussian process and to demonstrate that it meets strong Markov property are proved. As a result of strong Markov property, the reflection principle is shown and then applied repeatedly to obtain the proof of Lévy’s triple law. The invariance principle known as Donsker’s theorem is stated. Finally, Brownian bridge and its properties are defined, and it is proved that Brownian bridge is approximately Brownian motion conditioned to be in the neighborhood of the zero at time 1. At the end of this chapter we link Brownian bridge with empirical distribution function. The last chapter of this thesis introduces concepts related to testing of statistical hypotheses, then talks about Kolmogorov-Smirnov statistics defined by the empirical distribution function and displays statistical test in which it is used as a test statistic. Finally we come to the main result which tells us that the Kolmogorov-Smirnov statistic has an asymptotic distribution and that Brownian bridge, and thus Brownian motion give us a way to compute it.

• Keywords: Brownov most; Kolmogorov-Smirnovljeva statistika; vjerojatnosni prostor; slučajna varijabla; empirijska funkcija distribucije; Brownovo gibanje; Gaussovski proces; Lévyev zakon trojke; Donskerov teorem
• Keywords (english): Brownian bridge; Kolmogorov-Smirnov statistic; probability space; random variable; empirical distribution function; Brownian motion; Gaussian process; Lévy’s triple law; Donsker's theorem
• Language: croatian
• URN:NBN: urn:nbn:hr:217:258241
• Study programme: Title: Finance and Business Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2017-06-06 10:42:04