Najmanjši interval zaupanja za delež : delo diplomskega seminarja

Sokolič, Andrej (author), Smrekar, Jaka (mentor)

Ugotavljanje "prave" vrednosti parametra je zelo pomembno področje v sklepni statistiki. Zanimivo je, ker v splošnem "prave" vrednosti ne poznamo, lahko pa jo ocenimo. Cilj je najti način, ki bo "pravo" vrednost čim bolje ocenil. Eden najosnovnejših in najstarejših problemov je ocenjevanje deleža v populaciji. Čeprav se problem zdi zelo preprost, se izkaže, da to ni tako. Prav zato je, kljub temu da je na to temo napisanega že ogromno gradiva, še vedno aktualna tema. Za ocenjevanje deleža se kot zelo primerni izkažejo intervali zaupanja. V primerjavi z točkovnimi ocenami so uporabni tudi pri večkratnem(ponovnem) vzorčenju. Problema se lotimo tako, da določimo dva razreda enostranskih in razred dvostranskih intervalov zaupanja 1-alfa. Za intervale iz razredov zahtevamo monotonost v mejah zaupanja in simetrijo glede na verjetnost uspeha, parametra pri binomski porazdelitvi. V obeh enostranskih razredih iščemo najmanjši interval v smislu preseka vseh intervalov iz razreda z direktno analizo funkcije verjetnosti pokritosti. Ugotovimo, da s tako metodo dobimo tradicionalni Clopper-Pearsonov enostranski interval zaupanja 1-alfa, ki je v tem primeru tudi najmanjši. V razredu dvostranskih intervalov zaupanja najmanjši interval obstaja le, če je izpolnjen preprost pogoj, ki ga navedemo. Če je pogoj izpolnjen, najmanjši interval tudi izpeljemo. Predlagani intervali so enakomerno najbolj natančni in imajo enakomerno najmanjšo pričakovano dolžino.

• Language: Slovenian
• Keywords: interval zaupanja, binomska porazdelitev, verjetnost pokritosti
• Work type: Final seminar paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2019
• PID: 20.500.12556/RUL-109338
• UDC: 519.2
• COBISS.SI-ID: 18710105
• Publication date in RUL: 30.08.2019

SOKOLIČ, Andrej, 2019, Najmanjši interval zaupanja za delež : delo diplomskega seminarja [online]. Bachelor’s thesis. [Accessed 17 May 2025]. Retrieved from: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=eng&id=109338

Secondary language

• Language: English
• Title: Smallest confidence intervals for one binomial proportion

Abstract

One of the main focuses of study in inferential statistics is determining the "true" value of a parameter. It is interesting because generally the actual value for a parameter is never known. However, with statistical methods, it is possible to get an estimation for a parameter. Finding the population proportion is one of the oldest and most basic problems. Although the problem may seem easy at first, it soon becomes clear that it is not. Because of that, even after a lot of research, it is still a much discussed topic. Finding an interval estimate seems a reasonable method for estimating a proportion because especially under re-sampling, the point estimate can be very inaccurate. We start by defining two classes of one-sided and a class of two-sided 1-alpha confidence intervals with certain monotonicity and symmetry on the confidence limits for the probability of success, the parameter in a binomial distribution. In the class of one-sided confidence intervals we search for the smallest interval, in the sense of set inclusion, with direct analysis of the coverage probability function. This method gives us the same interval as the traditional one-sided 1-alpha Clopper-Pearson interval, which is in fact, although rarely mentioned, the smallest interval in the specified class. We provide a simple sufficient and neccessary condition for the existence of the smallest interval in the class of two-sided intervals. A method is provided for deriving the smallest interval if the condition holds. The proposed confidence intervals are uniformly most accurate and have the uniformly minimum expected length.

• Keywords: confidence interval, binomial distribution, coverage probability