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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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Testing Structural Changes Using Ratio Type Statistics
Peštová, Barbora ; Hušková, Marie (advisor) ; Prášková, Zuzana (referee) ; Jarušková, Daniela (referee)
Testing Structural Changes Using Ratio Type Statistics Barbora Peštová Charles University in Prague, Faculty of Mathematics and Physics, Department of Probability and Mathematical Statistics, Czech Republic Abstract of the doctoral thesis We deal with sequences of observations that are naturally ordered in time and assume various underlying stochastic models. These models are parametric and some of the parameters are possibly subject to change at some unknown time point. The main goal of this thesis is to test whether such an unknown change has occurred or not. The core of the change point methods presented here is in ratio type statistics based on maxima of cumulative sums. Firstly, an overview of thesis' starting points is given. Then we focus on methods for detecting a gradual change in mean. Consequently, procedures for detection of an abrupt change in mean are generalized by considering a score function. We explore the possibility of applying the bootstrap methods for obtaining critical values, while disturbances of the change point model are considered as weakly dependent. Procedures for detection of changes in parameters of linear regression models are shown as well and a permutation version of the test is derived. Then, a related problem of testing a change in autoregression parameter is studied....
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Zero inflated Poisson model
Veselý, Martin ; Komárek, Arnošt (advisor) ; Hlávka, Zdeněk (referee)
This paper deals with the zero-inflated Poisson distribution. First the Poisson model is defined and generalized to a zero-inflated model. The basic properties of this generalized model are derived. After- wards the basics of the method of moments and the maximum likelihood method are described. Both of these are used to derive parameter estimates of such distribution. The feasibility of calculating the distribution of moment method estimates is analyzed. Then the asymptotic distribution of maximum likelihood estimates is derived and used to create confidence intervals. In the last chapter a numeric si- mulation of the derived asymptotic properties is performed. Special attention is paid to situations where regularity conditions are not met. 1
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Testing Structural Changes Using Ratio Type Statistics
Peštová, Barbora ; Hušková, Marie (advisor) ; Prášková, Zuzana (referee) ; Jarušková, Daniela (referee)
Testing Structural Changes Using Ratio Type Statistics Barbora Peštová Charles University in Prague, Faculty of Mathematics and Physics, Department of Probability and Mathematical Statistics, Czech Republic Abstract of the doctoral thesis We deal with sequences of observations that are naturally ordered in time and assume various underlying stochastic models. These models are parametric and some of the parameters are possibly subject to change at some unknown time point. The main goal of this thesis is to test whether such an unknown change has occurred or not. The core of the change point methods presented here is in ratio type statistics based on maxima of cumulative sums. Firstly, an overview of thesis' starting points is given. Then we focus on methods for detecting a gradual change in mean. Consequently, procedures for detection of an abrupt change in mean are generalized by considering a score function. We explore the possibility of applying the bootstrap methods for obtaining critical values, while disturbances of the change point model are considered as weakly dependent. Procedures for detection of changes in parameters of linear regression models are shown as well and a permutation version of the test is derived. Then, a related problem of testing a change in autoregression parameter is studied....
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Stability in Autoregressive Time Series Models
Dvořák, Marek ; Prášková, Zuzana (advisor) ; Hušková, Marie (referee) ; Picek, Jan (referee)
The main subject of this thesis is a change point detection in stationary vector autoregressions. Various test statistics are proposed for the retrospective break point detection in the parameters of such models, in particular, the derivation of their asymptotic distribution under the null hypothesis of no change. Testing procedures are based on the maximum like- lihood principle and are derived under normality, nevertheless the asymptotic results are valid for broader class of distributions and involve also the models with certain form of dependence. Simulation studies document the quality of the results.
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Analysis of variance when the assumption of homoscedasticity is violated
Zavadilová, Anna ; Omelka, Marek (advisor) ; Komárek, Arnošt (referee)
The method known as analysis of variance of simple sort offers a possibility of how to test equality of mean values of several random selections. At the same time, however, it requires the random selections to originate from normal distribution and to meet the condition of homoscedasticity, i.e. the requirement of identity of variances. The aim of this Thesis is to analyse consequences of violation of the assumptions of normality and homoscedasticity of the input data. The first part of the Thesis presents an overview of the course of the method based on the analysis of variance with standard assumptions. It is followed by the deriving of asymptotic distribution of test statistics, supposing the validity of the null hypothesis of identity of mean values in the case that neither the normality of input data nor the identity of variances is supposed. The findings are then applied to several special cases. The final part of the Thesis deals with a simulation study describing the influence of non-fulfilment of assumptions imposed on the test significance level. Powered by TCPDF (www.tcpdf.org)
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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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Pearsonův correlation coefficient and its use in statistical inference
Németh, Richard ; Omelka, Marek (advisor) ; Maciak, Matúš (referee)
The main objective of this thesis is to determine asymptotic distribution of sample correlation coefficient without the assumption of normal distribution and its effects on commonly used sta- tistical tests of independence and confidence intervals for correlation coefficient. The problem is solved by central limit theorem and delta method. We have shown that the commonly used statis- tical tests for independence in practice are valid, even without the assumption of normal distribu- tion. We have also derived more versions of statistical tests for independence of random variables and more versions of confidence intervals for correlation coefficient without the assumption of normality. In conclusion we have compared individual statistical tests and confidence intervals for specific multivariate distributions using simulations. 1
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