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Models for zero-inflated data
Matula, Dominik ; Kulich, Michal (advisor) ; Hlubinka, Daniel (referee)
The aim of this thesis is to provide a comprehensive overview of the main approaches to modeling data loaded with redundant zeros. There are three main subclasses of zero modified models (ZMM) described here - zero inflated models (the main focus lies on models of this subclass), zero truncated models and hurdle models. Models of each subclass are defined and then a construction of maximum likelihood estimates of regression coefficients is described. ZMM models are mostly based on Poisson or negative binomial type 2 distribution (NB2). In this work, author has extended the theory to ZIM models generally based on any discrete distributions of exponential type. There is described a construction of MLE of regression coefficients of theese models, too. Just few of present works are interested in ZIM models based on negative binomial type 1 distribution (NB1). This distribution is not of exponential type therefore a common method of MLE construction in ZIM models cannot be used here. In this work provides modification of this method using quasi-likelihood method. There are two simulation studies concluding the work. 1
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Statistical analysis of datasets with missing observations
Janoušková, Kateřina ; Omelka, Marek (advisor) ; Kulich, Michal (referee)
Mechanisms of missing data and methods of their treatment are de- scribed in this thesis. Three mechanisms are considered - MCAR, MAR, MNAR. Two simple methods using deletion of incomplete records are introduced and their properties and shortcomings are described. Further, the principle of simple imputations is explained. EM algorithm which uses the classical statistics and the algorithm of data augmentation based on Bayesian framework are derived and compared. The last method included in the thesis is the multiple imputation. The described methods are applied on real data set, first on continuous variables and then on a two dimensional contingency table. 1
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GOF tests for gamma distribution
Klička, Petr ; Hlávka, Zdeněk (advisor) ; Kulich, Michal (referee)
The Bachelor thesis deals with the goodness of fit test for the Gamma distribution. Initially, we show several ways how to estimate the parameters of the Gamma distribution - firstly, the maximum likelihood estimator is presented, followed by estimator gained by the method of moments and fi- nally, we introduce the new estimator based on the sample covariance. The last estimator is used for constructing the goodness of fit test for the Gamma distribution. We define the test statistics V ∗ n to this test and its asymptotic normality is derived under the assumption of the null hypothesis. At the end of the thesis the simulations are realized to obtain the empirical size of the test for various values of parameter a and parameter b which equals one. 1
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Confidence Intervals for Binomial Parameters
Rusá, Pavla ; Kulich, Michal (advisor) ; Maciak, Matúš (referee)
The Bachelor thesis deals with the construction of confidence intervals for the parameter of the Binomial distribution. In the first part of the thesis we deal with the relationship between hypotheses testing and confidence in- tervals. The methods mentioned in this thesis are based on this relationship. The next part is devoted to the Clopper-Pearson method and its possible improvements which were made by Sterne, Crow, Blyth and Still. The gra- phical approach of Schilling and Doi is also worth noticing. Afterwards we present the methods based on the Normal approximation, particularly the Wilson method and the Wald method. Finally, all the methods mentioned in this thesis are compared in terms of coverage probability. 1
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Variable selection based on penalized likelihood
Chlubnová, Tereza ; Kulich, Michal (advisor) ; Maciak, Matúš (referee)
Selection of variables and estimation of regression coefficients in datasets with the number of variables exceeding the number of observations consti- tutes an often discussed topic in modern statistics. Today the maximum penalized likelihood method with an appropriately selected function of the parameter as the penalty is used for solving this problem. The penalty should evaluate the benefit of the variable and possibly mitigate or nullify the re- spective regression coefficient. The SCAD and LASSO penalty functions are popular for their ability to choose appropriate regressors and at the same time estimate the parameters in a model. This thesis presents an overview of up to date results in the area of characteristics of estimates obtained by using these two methods for both small number of regressors and multidimensional datasets in a normal linear model. Due to the fact that the amount of pe- nalty and therefore also the choice of the model is heavily influenced by the tuning parameter, this thesis further discusses its selection. The behavior of the LASSO and SCAD penalty functions for different values and possibili- ties for selection of the tuning parameter is tested with various numbers of regressors on simulated datasets.
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Models with categorical response
Faltýnková, Anežka ; Kalina, Jan (advisor) ; Kulich, Michal (referee)
This thesis concentrates on regression models with a categorical response. It focuses on the model of logistic regression with binary response and its generalization in which two models are distinguished: multinomial regression with nominal response and multinomial regression with ordinal response. For all three models separately, the Wald test and the likelihood ratio test are derived. These theoretical derivations are then used to calculate the test statistics for specific examples in statistical software R. The theory described in the thesis is illustrated by examples with small and large number of explanatory variables.
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Cure-rate models
Drabinová, Adéla ; Kulich, Michal (advisor) ; Omelka, Marek (referee)
In this work we deal with survival models, when we consider that with positive probability some patients never relapse because they are cured. We focus on two-component mixture model and model with biological motivation. For each model, we derive estimate of probability of cure and estimate of survival function of time to relaps of uncured patients by maximum likelihood method. Further we consider, that both probability of cure and survival time can depend on regressors. Models are then compared through simulation study. 1
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Multivariate Normal Distribution
Ježo, Jakub ; Kulich, Michal (advisor) ; Hlávka, Zdeněk (referee)
Title: Multivariate Normal Distribution Author: Jakub Ježo Department: Department of Probability and Mathematical Statistics Supervisor: doc. Mgr. Michal Kulich, Ph.D., Department of Probability and Mathematical Statistics Abstract: This bachelor thesis deals with the multivariate normal distribution, distributions derived from it and relations between them. The definition and characterization of the n-dimensional multinormal distribution, derivation of its characteristic function and definition of the matrix normal distribution are shown at the beginning. Further this thesis looks at the properties of the multivariate normal distribution and examines the linear combinations of normal vectors, li- near combinations of normal matrices and theirs properties. After that the qua- dratic forms of matrices from the normal distribution are shown, which leads to the Wishart distribution, its properties and the analysis of multidimensional data based on it. At the end of the thesis, the combinations of random vec- tors and matrix from the normal distribution are examined, which results to the Hotelling distribution and its properties. The distribution and properties of the sample mean vector and sample covariance matrix of a random sample from n- dimensional multinormal distribution are presented in this thesis....
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Levene's Test for Equality of Variances
Hrochová, Magdalena ; Omelka, Marek (advisor) ; Kulich, Michal (referee)
In the presented bachelor thesis, we focused on the Levene's test and its modifications that are used to assess the equality of variances for k independent random samples. At the beginning, we described Analysis of variance (ANOVA) that is a method for analyzing differences between group means for k independent random samples. The following part contains the original Levene's test description, including a discussion about the assumptions' verification for using ANOVA. Considering the fact that the original test is not confident in case of samples from specific distributions (e.g. a chi-square distribution) we summarize some known and suggest some new modifications. The main part of the thesis is dedicated to experimental simulations. In the conclusion, we discuss the simulation results for different versions of the Levene's test and other similar tests for equality of variances. 1
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Optimality of sample variance
Gleta, Filip ; Kulich, Michal (advisor) ; Anděl, Jiří (referee)
It is widely known that the most common estimators of the variance and the standard deviation based on i.i.d. data are not optimal with respect to the mean squared error. The aim of this thesis is to study and summarize the various approaches to seeking an improved estimator, which stem mainly from the innovative ideas presented by Stein (1964). Taken into consideration is the point estimator of the variance and the standard deviation. Each of the improved estimators include, in addition to their construction, a discussion regarding admissibility with respect to the MSE. Subsequently, using simple simulations for various distributions, it is examined whether obtained improvements lead to better results in practice. Powered by TCPDF (www.tcpdf.org)
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