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Priklausomų normaliųjų dydžių ekstremumų momentai / Moments of extremes of normally distributed valuesBurauskaitė, Agnė 09 June 2005 (has links)
Gaussian distribution is the most applied in practice and because of that reason there is a great amount of studies done in this area. In this report we look at Gaussian distribution from a point of view of extreme value theory. More concretely, moments of maximum of normally distributed values are discussed.
There are methods to calculate moments of extremes of independent identically distributed normal values, values with different variances and asymptotical results.
In this work a case of dependant variables is analyzed and aim is to look for results in similar cases that is done for independent variables. Continuing Bachelor’s work formula for moment calculation of maximum of two dependent normal variables with all different parameters is presented. Also there is a proof of formula for calculation of odd order moments of three dependent variable maximum. This result is generalized for random variable vectors of any length. There is a theorem stated, according to which moments of length n vector maximum could be expressed by same order moments of shorter vectors. Unfortunately, because of requirements for numbers n and m, no recursion method could be applied.
Using computer, maximum of various length random vectors with dependent components is simulated and average is analyzed. In experiments relation between mean values of dependent and independent variable maximums is observed. This relation is stated in a form of a formula and proved for vectors of any length. In this... [to full text]
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Daugiamačio pasiskirstymo tankio neparametrinis įvertinimas naudojant stebėjimų klasterizavimą / The nonparametric estimation of multivariate distribution density applying clustering proceduresRuzgas, Tomas 14 March 2007 (has links)
The paper is devoted to statistical nonparametric estimation of multivariate distribution density. The influence of data pre-clustering on the estimation accuracy of multimodal density is analysed by means of the Monte-Carlo method.
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Daugiamačio pasiskirstymo tankio neparametrinis įvertinimas naudojant stebėjimų klasterizavimą / The nonparametric estimation of multivariate distribution density applying clustering proceduresRuzgas, Tomas 15 March 2007 (has links)
The paper is devoted to statistical nonparametric estimation of multivariate distribution density. The influence of data pre-clustering on the estimation accuracy of multimodal density is analysed by means of the Monte-Carlo method.
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Netiesinių statistikų taikymas atsitiktinių vektorių pasiskirstymo tankių vertinime / Application of nonlinear statistics for distribution density estimation of random vectorsŠmidtaitė, Rasa 11 August 2008 (has links)
Statistikoje ir jos taikyme vienas dažniausiai sprendžiamų uždavinių yra daugiamačių tankių vertinimas.Tankių vertinimas skirstomas į parametrinį ir neparametrinį vertinimą. Parametriniame vertinime daroma prielaida, kad tankio funkcija f, apibūdinanti duomenis yi, kai i kinta nuo 1 iki n, priklauso tam tikrai gan siaurai funkcijų šeimai f(•;θ), kuri priklauso nuo nedidelio kiekio parametrų θ=(θ1, θ2, …, θk). Tankis, apskaičiuojamas pagal parametrinį vertinimą, gaunamas iš pradžių apskaičiavus parametro θ įvertį θ0 ir f0=f(•;θ). Toks traktavimas statistiniu požiūriu yra labai efektyvus, tačiau jeigu nei vienas šeimos f(•;θ) narys nėra artimas funkcijai f, rezultatai gali būti gauti labai netikslūs.
Neparametriniam tankio vertinimui jokios parametrinės prielaidos apie f nėra reikalingos, tačiau vietoj to daromos kitos prielaidos, pavyzdžiui, apie funkcijos f tolydumą arba, kad f yra integruojama. Tankio funkcijos forma yra nustatoma iš turimų duomenų.Turint dideles imtis, tankis f gali būti apskaičiuotas pakankamai tiksliai.
Šiuolaikinėje duomenų analizėje naudojama daugybė neparametrinių metodų, skirtų daugiamačių atsitiktinių dydžių pasiskirstymo tankio statistiniam vertinimui. Ypač plačiai paplitę branduoliniai įvertiniai, populiarūs ir splaininiai bei pusiau parametriniai algoritmai. Taikant daugumą populiarių neparametrinio įvertinimo procedūrų praktikoje susiduriama su jų parametrų optimalaus parinkimo problema. Branduolinių įvertinių konstrukcijos svarbiausiu... [toliau žr. visą tekstą] / Most algorithms work properly if the probability densities of the multivariate vectors are known. Unfortunately, in reality these densities are usually not available, and parametric or non-parametric estimation of the densities becomes critically needed.
In parametric estimation one assumes that the density f underlying the data yi where i varies from 1 to n, belongs to some rather restricted family of functions f(•;θ) indexed by a small number of parameters θ=(θ1, θ2, …, θk). An example is the family of multivariate normal densities which is parameterized by the mean vector and the covariance matrix. A density estimate in the parametric approach is obtained by computing from the data an estimate θ0 of θ and setting f0=f(•;θ). Such an approach is statistically and computationally very efficient but can lead poor results if none of the family members f(•;θ) is close to f.
In nonparametric density estimation no parametric assumptions about f are made and one assumes instead that f, for example, has some smoothness properties (e.g. two continuous derivatives) or that it is square integrable. The shape of the density estimate is determined by the data and, in principle, given enough data, arbitrary densities f can be estimated accurately. Most popular methods are the kernel estimator based on local smoothing of the data. Quite popular are histospline, semiparametric and projection pursuit algorithms. While constructing various probability density estimation methods the most... [to full text]
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