Simultani intervali zaupanja za več kvantilov neznane porazdelitve : delo diplomskega seminarja

Taškov, Nataša (author), Smrekar, Jaka (mentor)

Iskanje intervalov zaupanja je pomemben statistični problem, saj nam intervali zaupanja podajo neko intervalsko oceno iskane količine. Če imamo vzorec neodvisnih opazovanj neznane zvezne porazdelitve in nas zanima neki določen kvantil te porazdelitve, lahko za izbrani kvantil izračunamo interval zaupanja z določeno stopnjo zaupanja. Včasih pa si želimo izračunati intervale zaupanja za več kvantilov z neko določeno skupno stopnjo zaupanja. Temu pravimo simultani intervali zaupanja. Ker nimamo nobenih predpostavk o porazdelitvi, uporabimo neparametrične metode za izračun intervalov zaupanja. Iskanje intervalov zaupanja za en kvantil se da izračunati s pomočjo binomske metode. Metoda poišče intervale zaupanja s posamezno stopnjo zaupanja natanko $1-\alpha$. Če želimo poiskati intervale zaupanja, katerih skupna stopnja zaupanja je $1-\alpha$, potrebujemo drugačne metode. Dve od teh metod bosta predstavljeni v tem diplomskem delu. Če iščemo simultane intervale zaupanja za vse kvantile porazdelitve ali če nas zanimajo intervali zaupanja za veliko kvantilov, ki so razpršeni povsod po intervalu $(0,1)$, potem je najbolje pridobiti simultane intervale zaupanja iz pasov zaupanja. Eden od načinov iskanja pasov zaupanja je s pomočjo metode Kolmogorova. Simultani intervali zaupanja, ki jih dobimo iz pasov zaupanja so navadno zelo široki in zato pogosto neuporabni. Obstaja pa metoda, ki temelji na multinomski porazdelitvi, ki poišče simultane intervale zaupanja le za nekaj izbranih kvantilov. Ti intervali zaupanja so običajno ožji. Za izračun simultane stopnje zaupanja obstaja rekurziven algoritem, katerega računska zahtevnost raste s številom kvantilov, ki nas zanimajo. Za simultane intervale zaupanja želimo poiskati množice optimalnih mejnih vrednosti. Da bi jih lažje našli, si lahko pomagamo s petimi kriteriji: simterija, enostranskost in dvostranskost, razpon intervala zaupanja, stopnja zaupanja in gnezdenje. Simultani intervali zaupanja, ki jih dobimo z multinomsko metodo so običajno ožji kot tisti, ki jih dobimo iz pasov zaupanja, in širši od intervalov zaupanja za posamezne kvantile. Če iščemo simultane intervale zaupanja s pomočjo multinomske metode in simultane intervale zaupanja s pomočjo parametričnih metod, je bolje uporabiti neparametrične metode, saj tako intervali zaupanja niso odvisni od prepodstavk o porazdelitvi.

• Language: Slovenian
• Keywords: simultani intervali zaupanja, intervali zaupanja, pasovi zaupanja, binomska porazdelitev, multinomska porazdelitev, neparametrične metode, rekurzivni izračuni
• Work type: Final seminar paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2018
• PID: 20.500.12556/RUL-103473
• UDC: 519.2
• COBISS.SI-ID: 18440537
• Publication date in RUL: 19.09.2018

TAŠKOV, Nataša, 2018, Simultani intervali zaupanja za več kvantilov neznane porazdelitve : delo diplomskega seminarja [online]. Bachelor’s thesis. [Accessed 17 May 2025]. Retrieved from: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=eng&id=103473

Secondary language

• Language: English
• Title: Simultaneous confidence intervals for several quantiles of an unknown distribution

Abstract

Finding confidence intervals is an important statistical problem because the confidence intervals provide an interval evaluation of the quantity of interest. Supose that we are interested in a certain quantile of an unknown continuous distribution, given a sample of independent observations from this distribution, we can calculate a confidence interval with a certain confidence level for the particular quantile. Sometimes we want to calculate a confidence interval for several quantiles with a certain total confidence level. We call this a simultaneous confidence interval. Since we do not have any distribution assumptions, we use non-parametric methods for calculating confidence intervals. Searching for confidence intervals for a single quantile can be calculated using the binomial method. The method searches for confidence intervals with an individual confidence level exactly $1-\alpha$. If we want to search for confidence intervals whose total confidence level is $1-\alpha$, we need different methods. Two of these methods will be presented in this work. If we search for confidence intervals for all quantiles of an unknown distribution, or if we are interested in confidence intervals for many quantiles that are scattered over the entire interval $(0,1)$, then it is best to obtain confidence intervals from confidence bands. One of the ways to search for confidence bands is using the Kolmogorov method. Simultaneous confidence intervals obtained from confidence bands are usually very wide and therefore often useless. Therefore, there is a method based on multinomial distribution that searches for simultaneous confidence intervals for only a few selected quantiles. These confidence intervals are usually narrower. To calculate the simultaneous confidence level, there is a recursive algorithm that has a computational complexity that increases linearly with the number of quantiles that interest us. For simultaneous confidence intervals we want to find a set of super-feasible boundary values values. In order to find them, we employ five criteria: symmetry, one-sidedness and two-sidedness, confidence interval spread, confidence level and inclusion. Simultaneous confidence intervals obtained by the multinomial method are usually narrower than those obtained from confidence bands and wider than the confidence intervals for individual quantile. If we are looking for simultaneous confidence intervals by means of the multinomial method, it is preferable to use nonparametric methods, since then the confidence intervals do not depend on distribution assumptions.

• Keywords: simultaneous confidence intervals, confidence intervals, confidence bands, binomial distribution, multinomial distribution, nonparametric methods, recursive computations