Bootstrap metoda u linearnoj regresiji

master's thesis

• Title (english): Bootstrap method in linear regression
• Author: Eleonora Detić
• Mentor: Snježana Lubura Strunjak (mentor)
• Committee member: Snježana Lubura Strunjak (predsjednik povjerenstva)
• Committee member: Marcela Hanzer (član povjerenstva)
• Committee member: Vanja Wagner (član povjerenstva)
• Committee member: Nela Bosner (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2023-03-03, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Linearna regresija najčešći je oblik regresijske analize. Ako su pretpostavke normalnosti, homogenosti i nezavisnosti slučajnih grešaka ispunjene, metoda najmanjih kvadrata daje egzaktne rezultate. U praksi su pretpostavke često oslabljene ili čak nedostižne, čime se traži alternativa u procjeni koeficijenata. Metode ponovnog uzorkovanja koje se zasnivaju na zakonu velikih brojeva nameću se kao moguće potencijalno rješenje. U ovom radu proučava se bootstrap metoda, koja polazi od ideje da statistički uzorak već sadrži sve dostupne informacije o nekoj funkciji distribucije F te ponovnim uzorkovanjem dobiva se uzorak koji pripada toj istoj distribuciji. U središtu interesa su dva algoritma - metoda uzorkovanja podataka i metoda uzorkovanja grešaka. U prvom poglavlju iznose se osnovni pojmovi teorije vjerojatnosti te se predstavlja bootstrap metoda. U drugom poglavlju definiran je jednostavan model linearne regresije te je iznesena ideja bootstrap metode za distribuciju procjenitelja. Treće poglavlje je proširivanje drugog poglavlja na višedimenzionalan slučaj. Također, predstavljen je pojam predikcije te uloga bootstrap metode u tom pogledu. U četvrtom poglavlju izloženi su praktični rezultati dobiveni raznim simulacijama u programskom jeziku R. Dodatak na kraju rada je peto poglavlje, gdje se nalazi kod korišten za provođenje simulacija.

Abstract (english)

Linear regression is the most common form of regression analysis. If the assumptions of normality, homogeneity and independence of random errors are satisfied, the least square method gives exact results. In practice, assumptions are often difficult to reach or even elusive, which requires looking for an alternative. Resampling methods, based on the law of large numbers, are offered as a possible potential solution. In this paper, the bootstrap method is studied, which is based on the idea that a statistical sample already contains all available information about a distribution function F, so by resampling, we get a sample that belongs to that same distribution. Two algorithms are particularly discussed – the data resampling method and the errors resampling method. In the first chapter, the basic concepts of probability theory and the bootstrap method are presented. In the second chapter, a simple linear regression model is defined and the idea of the bootstrap method for the distribution of estimators is exposed. The third chapter is an extension of the second chapter to a multidimensional case. Also, the concept of prediction and the role of the bootstrap method in this respect are presented. In the fourth chapter, the practical results obtained by conducting various simulations in the R programming language are exposed. The code used to perform the simulations is presented at the end of the paper, in the fifth chapter.

• Keywords: regresijska analiza; metode ponovnog uzorkovanja; zakon velikih brojeva; metoda uzorkovanja podataka; metoda uzorkovanja grešaka; programski jezik R
• Keywords (english): regression analysis; resampling methods; law of large numbers; data sampling method; errors resampling method; R programming language
• Language: croatian
• URN:NBN: urn:nbn:hr:217:073765
• 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: 2023-03-23 15:34:53