Bootstrap metoda za pouzdane intervale

master's thesis

• Title (english): Bootstrap method for confidence intervals
• Author: Tea Maričić
• Mentor: Snježana Lubura Strunjak (mentor)
• Committee member: Snježana Lubura Strunjak (predsjednik povjerenstva)
• Committee member: Boris Širola (član povjerenstva)
• Committee member: Goran Radunović (član povjerenstva)
• Committee member: Goran Muić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2022-09-28, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom diplomskom radu ukratko smo opisali bootstrap metodu. Napravili smo jednostavan primjer parametarske i neparametarske simulacije te smo promotrili histograme aritmetičkih sredina. Objasnili smo načine na koje se pomoću nje može doći do procjene pouzdanih intervala. Opisali smo bootstrap t-metodu, percentilnu te \(\text{BC}_a\) metodu za procjenu pouzdanog intervala. Sve metode potkrijepili smo simulacijama za parametar očekivanja na slučajnom uzorku iz normalne distribucije. Vidjeli smo kako ovisno o metodi naši intervali mogu biti širi i uži te neovisno o parametarskoj i neparametarskoj procjeni stvarna vrijednost parametra uvijek ne mora upadati u dobiveni procijenjeni interval. \(\text{BC}_a\) metoda je većinom davala najuži interval dok je u većini slučajeva studentizirana ili bootstrap-t metoda davala najširi interval. Pogledali smo simulacije za očekivanje, medijan i varijancu iz raznih razdioba. U ovom radu simulaciju smo proveli za normalnu, eksponencijalnu i gamma razdiobu. Zaključili smo kako točnost naše procjene uvelike ovisi o samoj statistici, uzorku i distribuciji. Najbolji rezultati su za simetrične distribucije.

Abstract (english)

In this master thesis we presented bootstrap method. We made simple example of parametric and nonparametric simulation based on which we observed histogram of means. We explained methods where using bootstrap can be useful method to estimate confidence intervals for statistic of interest. We described bootstrap-t method, percentile and \(\text{BC}_a\) method for estimation of confidence interval. All methods were supported by simulations for mean confidence intervals on sample from normal distribution. We saw that depending on method our intervals can be more or less wide and independent from parametric or nonparametric simulation real value of parameter does not always have to be contained in estimated confidence interval. \(\text{BC}_a\) method mostly gave us the narrowest interval. On the other hand, student or bootstrap-t method in most of the cases gave us the widest interval for parameter of interest. In the last chapter we saw simulations for mean, median and variance on samples that came from diverse distributions. In this thesis we did simulation for normal, exponential and gamma distribution. We concluded that accuracy of our estimation mostly depends on statistic, sample and distribution. The best results were obtained for symmetrical distributions.

• Keywords: parametarska simulacija; neparametarska simulacija; histogrami aritmetičkih sredina; bootstrap t-metoda; percentilna metoda
• Keywords (english): parametric simulation; nonparametric simulation; histogram; bootstrap-t method; percentile method
• Language: croatian
• URN:NBN: urn:nbn:hr:217:021699
• 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: 2022-10-31 12:21:45