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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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Analysis of Outcome Change in Randomized Studies
Hanuš, Antonín ; Kulich, Michal (advisor) ; Malý, Marek (referee)
Antonín Hanuš 5. prosince 2014 This work deals with randomized clinical trials of medicaments. It examines three models of dependece of final values on initial values in case, that all variables are measured with some measurement error. For each model is derived effect of treatment estimate and its asymptotical properties, specifically consistency and asymptotical variance. The work mostly deals with linear model of analysis of covariance ANCOVA. The work fruther contents comparison of properties of estimates from all three models in case, that examined data come from a linear model. There is a comparison of asymptotical variances of estimates from all three models and for each of them there are examined conditions, when this model gives the best results. In the end there is a simulation study, which verifies all previous results. 1
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Ověřování předpokladů modelu proporcionálního rizika
Marčiny, Jakub ; Kulich, Michal (advisor) ; Zvára, Karel (referee)
The Cox proportional hazards model is a standard tool for modelling the effect of covariates on time to event in the presence of censoring. The appropriateness of this model is conditioned by the validity of the proportional hazards assumption. The assumption is explained in the thesis and methods for its testing are described in detail. The tests are implemented in R, including self-written version of the Lin- Zhang-Davidian test. Their application is illustrated on medical data. The ability of the tests to reveal the violation of the proportional hazards assumption is investigated in a simulation study. The results suggest that the highest power is attained by the newly implemented Lin-Zhang-Davidian test in most cases. In contrast, the weighted version of the Lin-Wei-Ying test was found to have inadequate size for low sample sizes.
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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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Least Squares Alternatives
Gerthofer, Michal ; Pešta, Michal (advisor) ; Kulich, Michal (referee)
In the present thesis we deal with the linear regression models based on least squares. These methods are discussed in two groups. The first one focuses on three primary aproaches devided by occurrence of errors in variables. The traditional approach penalizes only the misfit in the de- pendent variable part and is called the ordinary least squares (OLS). An opposite case to the OLS is represented by the data least squares (DLS), which allow corrections only in the explanatory variables. Consecutively, we concentrate ourselves on the total least squares approach (TLS) mi- nimizing the squares of errors in the values of both dependent and independent variables. Finally, we give attention to next group of methods whit high breakdown point, which deal with signifi- cance of the individual observations (least weighted squares) and elimination of outlying obser- vations (least trimmed squares). The main purpose of this work is to describe and compare these models, their assumptions, characteristics, properties of estimates and show them on real data. 1
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Basic Multivariate Distributions
Sýkorová, Sabina ; Kulich, Michal (advisor) ; Hurt, Jan (referee)
The thesis deals with the basic discrete and continuous multivariate distributions, which play an important role in statistical analyses of models in applied fields. It focuses mainly on the derivation of these distributions using various techniques by which univariate distributions are generalized to higher dimensions. At the beginning of the thesis the multivariate normal distribution is defined, than it deals with distributions that are derived by direct generalization of univariate distributions. These are multivariate log-normal, multivariate Student's, multivariate Pareto, Dirichlet, and multinomial distributions. Furthermore it describes a common components method by which a multivariate Poisson distribution and a multivariate gamma distribution are derived. In the last chapter we introduce a multivariate exponential distribution derived by a stochastic generalization technique.
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Group sequential tests in clinical trials
Jílek, Josef ; Kulich, Michal (advisor) ; Komárek, Arnošt (referee)
Group sequential tests are an important statistical method. The analysis of data are performed continuously, which allows us to terminate the test before all observations are collected. For example these tests are used in medicine. When testing new drugs or procedures, this method brings about financial savings as well as ethical advantages. There are many ways of conducting group sequential tests with different qualities. Based on the perused literature, both basic and more complex types of group sequential tests are introduced in this paper. It discribes their principle and respective examples are provided. With this information it is possible to design and conduct a particular test. It's merits and demerits are compared for every method in real situations. The result is a tabular scale of different tests, from which it is possible to select a particular test for a given situation.
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Applications of EM-algorithm
Komora, Antonín ; Omelka, Marek (advisor) ; Kulich, Michal (referee)
EM algorithm is a very valuable tool in solving statistical problems, where the data presented is incomplete. It is an iterative algorithm, which in its first step estimates the missing data based on the parameter estimate from the last iteration and the given data and it does so by using the conditional expectation. In the second step it uses the maximum likelihood estimation to find the value that maximizes the logarithmic likelihood function and passes it along to the next iteration. This is repeated until the point, where the value increment of the logarithmic likelihood function is small enough to stop the algorithm without significant errors. A very important characteristic of this algorithm is its monotone convergence and that it does so under fairly general conditions. However the convergence itself is not very fast, and therefore at times requires a great number of iterations.
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Introduction to Linear Mixed Models
Šaroch, Vojtěch ; Kulich, Michal (advisor) ; Komárek, Arnošt (referee)
of the bachelor thesis Title: Introduction to Linear Mixed Models Author: Vojtěch Šaroch Department: Department of Probability and Mathematical Statistics, MFF UK Supervisor: doc. Mgr. Michal Kulich Ph.D. Abstract: The thesis describes general procedures of estimation and hypothesis testing for linear statistical models. The models compare groups of observation due to dependent variable. Analysis of variance and linar mixed models are commonly used in the major science like pharmacology, biochemistry, economy and others. The thesis is appropriate for general public, because no advanced knowledge of probability and statistics are required. Particular methods are introduced gently and contain some practical examples for easier understanding of theory. Keywords: Analysis of variance (ANOVA), fixed and random effect, linear mixed model 1
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Poisson Approximations
Klikáč, Jan ; Omelka, Marek (advisor) ; Kulich, Michal (referee)
This bachelor thesis deals with the counting probability using Poisson distri- bution and shows new ways of approximating Poisson distribution. The first chapter summarizes the findings regarding the Poisson distribution, its definition and properties. It also show a limit transition from the binomial distribution to Poisson distibution and examples demonstrating the usage of this limit transition. Brun Sieve is introduced in the second chapter. It gives a new possibility of transiting to a Poisson distribution. Random variables, which we want to appro- ximate, no longer need to have binomial distribution. Instead the property of expected value is required. The second part of the chapter includes a practical demonstration of the usage of Brun Sieve. In the third chapter we estimate size of the error that we made when approxi- mating to Poisson distribution. There is also formulated Stein-Chen theorem for estimating the error of Poisson approximation and its version for a special case. Keywords: Poisson distribution, Brun Sieve, Stein-Chen theorem 1
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