Metoda bootstrapa u statističkom zaključivanju

master's thesis

• Author: Ivan Vajdić
• Mentor: Miljenko Huzak (mentor)
• Committee member: Miljenko Huzak (predsjednik povjerenstva)
• Committee member: Bojan Basrak (član povjerenstva)
• Committee member: Snježana Lubura Strunjak (član povjerenstva)
• Committee member: Vanja Wagner (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2020-07-20, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovome radu ilustrirali smo korištenje metode bootstrapa u svrhu testiranja statističkih hipoteza i konstrukcije pouzdanih intervala za parametre raznih modela. Iako su za obje ove metode statističkog zaključivanja izvedeni mnogobrojni teorijski rezultati, u praksi često pretpostavke pod kojima ti rezultati vrijede nisu ispunjeni. Tada je potrebno posegnuti za aproksimativnim metodama. Glavna je ideja ove metode procijeniti određene parametre iz uzorka i koristiti te procjene kao da su pravi parametri. Intuitivno, sva informacija o populaciji sadržana je u uzorku pa ovim postupkom tretiramo uzorak kao populaciju. Simuliranjem iz procijenjene razdiobe dobivamo ideju kako izgleda uzoračka razdioba neke statistike od interesa. Iako je metoda aproksimativna, kroz mnogobrojne primjere pokazali smo da daje zadovoljavajuće dobre rezultate te brzu i direktnu procjenu. Pokazali smo i kako se svaka od tih metoda može dodatno korigirati ne bi li rezultati bili još precizniji. Razvili smo metode koje iznenađujuće dobro funkcioniraju i u slučajevima kad nemamo nikakvih pretpostavki na razdiobu iz koje dolaze naša opažanja. Naravno, ako imamo razuman pretpostavljeni model, rezultati su još i bolji. Svaku predloženu metodu ilustrirali smo primjerima gdje smo koristili simulacije na računalu koje znatno olakšavaju izvrednjavanje kompliciranih izraza (na primjer, raznih vjerojatnosti) na koje se te metode oslanjaju

Abstract (english)

In this thesis we’ve illustrated the use of the bootstrap method for testing statistical hypotheses and constructing confidence intervals for parameters of various models. Even though there is a great number of theoretical results derived for both of these methods of statistical inference, often the assumptions under which those results work are not met in practice. This is the reason why we need to resort to approximate methods. The main point of this method is to use the available sample to estimate certain parameters and then use those estimates as if they were the real parameter values. Intuitively, all the information about the population of interest is contained in the sample, so basically we treat the sample as the population. By simulating samples from the fitted distribution we get an insight on the shape of sampling distribution of statistic of interest. Although this method is only approximate, we’ve shown through numerous examples how efficient it can be and that it results in satisfying and quick estimates. We’ve also demonstrated how each of the methods we presented can be improved so that they give adjusted and even more precise results. We’ve developed various methods that work surprisingly well even when no assumptions are being made on the distribution of the sample values. Certainly the results are even more accurate when we have a parametric model available. Every method that we’ve presented was accompanied by an example illustrating the use of the method. In the examples we used computer simulation to help us compute complicated expressions (for instance, some complicated probabilities) which are used in the methods.

• Keywords: metoda bootstrapa; statističke hipoteze
• Keywords (english): bootstrap method; statistical hypotheses
• Language: croatian
• URN:NBN: urn:nbn:hr:217:744511
• 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: 2020-11-30 13:29:46