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Copula-based multivariate association measures and tail coefficients
Kika, Vojtěch ; Omelka, Marek (advisor) ; Veraverbeke, Noel (referee) ; Fuchs, Sebastian (referee)
The dependence structure of a d-variate random vector X is a very complex notion which is fully described by the distribution of the random vector. Alternatively, it suffices to look into the corresponding copula function of X, as it ignores the marginal distributions of X but still fully describes the dependence structure. However, a copula is a function defined on the d-dimensional hypercube [0, 1]d with values in the interval [0, 1]. As such, it might be too complex for practical use and one would prefer to have tools that can translate the information from the copula function into a simpler indicator. In particular, of interest might be an association measure, that is, a single number that describes the tendency of the components of X to simultaneously take large or small values. Coefficients like Kendall's tau or Spearman's rho, used to measure (strength of) an association between two random variables, were thoroughly studied and described in the middle of 20th century. Requirements on bivariate association measures are well-known. However, generalization of such measures into higher dimensions is not very straightforward and brings discussion on the desirable properties. In addition, bivariate association measures can be often generalized in multiple manners. The same holds true if one wants to...
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Analysis of variance when the assumption of normality is violated
Kika, Vojtěch ; Omelka, Marek (advisor) ; Kulich, Michal (referee)
Attention is restricted to a method called Analysis of variance (ANOVA) that is used to compare expected values of several independent random samples. The clas- sic ANOVA theory with all its assumptions, including the assumption of normality, is presented at the beginning. Afterwards, an instance when the assumption of nor- mality of input data is violated is exemplified. The asymptotic distribution of test statistic under the hypothesis of the equality of the expected values is derived. The distribution is used to test the equality. Subsequently, it is shown that Tukey's range test and Scheffé's method of multiple comparison in case of non-normality could be used in the same way as for normal samples. The methods serve for compa- ring expected values of pairs of random samples. Thus, they can determine expected values which are different. Finally, a simulation study is presented which is to verify the proved theoretical results and to describe situations with data from non-normal distributions.
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Statistical inference in multivariate distributions based on copula models
Kika, Vojtěch ; Omelka, Marek (advisor) ; Hlubinka, Daniel (referee)
Diploma thesis abstract Thesis title: Statistical inference in multivariate distributions based on copula models Author: Vojtěch Kika This diploma thesis aims for statistical inference in copula based models. Ba- sics of copula theory are described, followed by methods for statistical inference. These are divided into three main groups. First of them are parametric methods for copula parameter estimation which assume fully parametric structure, thus for both joint and marginal distributions. The second group consists of semi- parametric methods for copula parameter estimation which, unlike parametric methods, do not require parametric structure for marginal distributions. The last group describes goodness-of-fit tests used for testing the hypothesis that consi- dered copula belongs to some specific copula family. The thesis is accompanied by a simulation study that investigates the dependence of the observed coverage of the asymptotic confidence intervals for copula parameter on the sample size. Pseudolikelihood method was chosen for the simulation study since it is one of the most popular semiparametric methods. It is shown that sample size of 50 seems to be sufficient for the observed coverage to be close to the theoretical one. For Frank and Gumbel-Hougaard copula families even sample size of 30 gives us...
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Analysis of variance when the assumption of normality is violated
Kika, Vojtěch ; Omelka, Marek (advisor) ; Kulich, Michal (referee)
Attention is restricted to a method called Analysis of variance (ANOVA) that is used to compare expected values of several independent random samples. The clas- sic ANOVA theory with all its assumptions, including the assumption of normality, is presented at the beginning. Afterwards, an instance when the assumption of nor- mality of input data is violated is exemplified. The asymptotic distribution of test statistic under the hypothesis of the equality of the expected values is derived. The distribution is used to test the equality. Subsequently, it is shown that Tukey's range test and Scheffé's method of multiple comparison in case of non-normality could be used in the same way as for normal samples. The methods serve for compa- ring expected values of pairs of random samples. Thus, they can determine expected values which are different. Finally, a simulation study is presented which is to verify the proved theoretical results and to describe situations with data from non-normal distributions.
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