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Statistical analysis of samples from the generalized exponential distribution
Votavová, Helena ; Popela, Pavel (referee) ; Michálek, Jaroslav (advisor)
Diplomová práce se zabývá zobecněným exponenciálním rozdělením jako alternativou k Weibullovu a log-normálnímu rozdělení. Jsou popsány základní charakteristiky tohoto rozdělení a metody odhadu parametrů. Samostatná kapitola je věnována testům dobré shody. Druhá část práce se zabývá cenzorovanými výběry. Jsou uvedeny ukázkové příklady pro exponenciální rozdělení. Dále je studován případ cenzorování typu I zleva, který dosud nebyl publikován. Pro tento speciální případ jsou provedeny simulace s podrobným popisem vlastností a chování. Dále je pro toto rozdělení odvozen EM algoritmus a jeho efektivita je porovnána s metodou maximální věrohodnosti. Vypracovaná teorie je aplikována pro analýzu environmentálních dat.
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Statistical Analysis of Extreme Value Distributions for Censored Data
Chabičovský, Martin ; Karpíšek, Zdeněk (referee) ; Michálek, Jaroslav (advisor)
The thesis deals with extreme value distributions and censored samples. Theoretical part describes a maximum likelihood method, types of censored samples and introduce a extreme value distributions. In the thesis are derived likelihood equations for censored samples from exponential, Weibull, lognormal, Gumbel and generalized extreme value distribution. For these distributions are also derived asymptotic interval estimates and is made simulation studies on the dependence of the parameter estimate on the percentage of censoring.
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Modeling of scoring probability in sport
Hilscher, Ondřej ; Bednář, Josef (referee) ; Hrabec, Pavel (advisor)
This thesis aims for modelling of scoring probability in football. It describes necessary mathematical methods used in logistic regression model building and in basic statistical hypothesis tests. Afterwards the mathematical methods are used on available data from professional football matches. Resulting model uses shooting method, pitch location and simplified match situation as predictors.
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Credit crunch v modelu nerovnováhy na peněžním trhu v České republice
Režňáková, Lucie
This diploma thesis deals with the credit crunch in the model of disequilibrium in the credit market. The basis of the empirical analysis is the application of methods of maximum likelihood on the modified time series. Using this method will be by estimated each function of supply and demand, on the basis of which will be determined by the individual disequilibrium. The results from the overall analysis will help us make recommendations for policy-makers
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Modeling of scoring probability in sport
Hilscher, Ondřej ; Bednář, Josef (referee) ; Hrabec, Pavel (advisor)
This thesis aims for modelling of scoring probability in football. It describes necessary mathematical methods used in logistic regression model building and in basic statistical hypothesis tests. Afterwards the mathematical methods are used on available data from professional football matches. Resulting model uses shooting method, pitch location and simplified match situation as predictors.
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Gini coefficient maximization in binary logistic regression
Říha, Samuel ; Hanzák, Tomáš (advisor) ; Hlávka, Zdeněk (referee)
This Bachelor thesis describes a binary logistic regression model. By means of the term loss function a parameter estimation for the model is derived. A "rich" set of "proper" loss functions - beta family of Fisher-consistent loss functions - is defined. In the second part of the thesis, four basic goodness-of-fit criteria - Gini coefficient, C-statistics, Kolmogorov-Smirnov statistics and coefficient of determination R2 are defined. Further on, a possibility of parameter estimation by maximizing the Gini coefficient is analysed. Several algorithms are designed for this purpose. They are compared with so far existing methods in one simulated data set and three real ones. 1
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Statistical inference for Markov processes with continuous time
Křepinská, Dana ; Prokešová, Michaela (advisor) ; Lachout, Petr (referee)
Tato diplomová práce se zabývá odhadováním matice intenzit Markovova pro- cesu se spojitým časem na základě diskrétně pozorovaných dat. Začátek práce je věnován jednoduššímu odhadu ze spojité trajektorie pomocí metody maximální věrohodnosti. Dále je zde popsán odhad z diskrétní trajektorie přes výpočet ma- tice pravděpodobností přechodu. Následně je velmi podrobně rozebrán EM al- goritmus, který předchozí odhad zpřesňuje. Na závěr teoretické části je uvedena metoda odhadu zvaná Monte Carlo Markov Chain. Všechny postupy jsou zároveň implementovány v počítačovém softwaru a prezentace jejich výsledk· je obsahem druhé části práce. V té jsou porovnané odhady pro denní, týdenní a měsíční po- zorování a také pro pětiletou a desetiletou pozorovanou trajektorii. K výsledk·m jsou připojeny odhady rozptyl· a intervaly spolehlivosti. 1
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Parameter estimation of random variables distribution
Šimková, Barbora ; Mošna, František (advisor) ; Novotná, Jarmila (referee)
of the bachelor's thesis Title: Parameter estimation of random variables distribution Author: Bc. Barbora Šimková Department: Department of Mathematics and Mathematical Education Supervisor: RNDr. František Mošna, Dr. Abstract: The subject of this thesis is to compare basic methods by which it is possible to calculate point estimates of discrete and continuous probability distributions. The work deals with the analysis of the two methods - the method of moments and maximum like- lihood method. These methods are used for point estimates of probability distributions parameters. The method of moments studies the comparison between the theoretical and sample moments of a random variable. The method of maximum likelihood is another alternative for the calculation of point estimates, which uses the classical ap- proach of finding the maximum of a function, using the properties of random selection. These methods of calculation are based on statistical methods and could be useful for extending the basic course on probability and statistics at Charles University's Fac- ulty of Education. The work is an overview of the estimated parameters of the basic distribution and compares the quality of two basic methods for their estimation. Keywords: parameter estimation, distribution of random variables, maximum...
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Estimation in continuous time Markov chains
Nemčovič, Bohuš ; Prokešová, Michaela (advisor) ; Kadlec, Karel (referee)
Title: Estimation in continuous time Markov chains Author: Bohuš Nemčovič Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Michaela Prokešová, Ph.D., Department of Probability and Mathematical Statistics Abstract: In this work we deal with estimating the intensity matrices of continu- ous Markov chains in the case of complete observation and observation at selected discrete time points. To obtain an estimate we use the maximum likelihood met- hod. In the second chapter we first introduce the general EM algorithm and then adjust it for finding the intensity matrix estimate based on observations at disc- rete time points. In the last chapter we will illustrate the impact of the discrete step size on the quality of intensity matrix estimate. Keywords: Markov chains, intensity matrix, maximum likelihood estimation, EM algorithm 1
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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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