Asimptotika procjenitelja maksimalne vjerodostojnosti

master's thesis

• Title (english): Asymptotics of the maximum plausibility estimator
• Author: Marin Krtinić
• Mentor: Miljenko Huzak (mentor)
• Committee member: Miljenko Huzak (predsjednik povjerenstva)
• Committee member: Vjekoslav Kovač (član povjerenstva)
• Committee member: Snježana Lubura Strunjak (član povjerenstva)
• Committee member: Zrinka Franušić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2021-09-22, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu se bavimo asimptotskim svojstvima niza procjenitelja maksimalne vjerodostojnosti. Na početku uvodimo osnovne pojmove matematičke statistike te navodimo rezultate iz teorije vjerojatnosti koje koristimo u ostatku rada. U drugom poglavlju iskazujemo i dokazujemo jednu verziju uniformnog jakog zakona velikih brojeva. Glavni rezultat rada je konzistentnost i asimptotska normalnost niza procjenitelja maksimalne vjerodostojnosti i obrađen je kroz treće i četvrto poglavlje. Uz to u navedenim poglavljima ilustriramo potencijalne probleme procjenitelja maksimalne vjerodostojnosti, ali i lijepo ponašanje na velikoj klasi često korištenih modela, kao na primjer u jednoparametarskoj eksponencijalnoj familiji. Na kraju uvodimo pojam asimptotske efikasnosti i pokazujemo kako se uz određene pretpostavke može iz konzistentnog i asimptotski normalnog niza procjenitelja konstruirati asimptotski efikasan niz procjenitelja.

Abstract (english)

In this thesis, we deal with the asymptotic properties of a sequence of maximum likelihood estimators. At the beginning, we introduce the basic concepts of mathematical statistics and list the results from probability theory that we use in the rest of the thesis. In the second chapter we state and prove one version of the uniform strong law of large numbers. The main result of the thesis is the consistency and asymptotic normality of a sequence of maximum likelihood estimators and it is processed through the third and fourth chapters. In addition, in the above chapters we illustrate the potential problems of the maximum likelihood estimator, but also the nice behavior on a large class of frequently used models, such as in a one-parameter exponential family. Finally, we introduce the notion of asymptotic efficiency and show how, under certain assumptions, an asymptotically efficient sequence of estimators can be constructed from a consistent and asymptotically normal sequence of estimators.

• Keywords: matematička statistika; teorija vjerojatnosti; zakona velikih brojeva; jednoparametarska eksponencijalna familija; asimptotska efikasnost
• Keywords (english): mathematical statistics; probability theory; law of large numbers; one-parameter exponential family; asymptotic efficiency
• Language: croatian
• URN:NBN: urn:nbn:hr:217:987183
• Study programme: Title: Mathematical Statistics; 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: 2021-10-28 11:50:51