|
|
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 an- other alternative for the calculation of point estimates, which uses the classical approach 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 used as an inter- isting extencion of the basic course on probability and statistics at Charles University's Faculty 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...
|
|
|
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...
|
|
|
Estimation of the survival function in the reliability analysis
Vojtěch, Jonáš ; Novák, Petr (advisor) ; Hurt, Jan (referee)
Present Bachelor thesis deals with the basic concepts and methods used in the survival analysis. Both nonparametric and parametric approaches to the estimation of the survival function are described. Nonparametric Kaplan Meier method is presented in order to estimate the survival function and consequently derive its basic properties. From the point of the probability distributions used in the analysis of reliability, exponential, Weibull's and logarithmic-normal distri- butions are applied. Parameters in the parametric approach to the estimation of the survival function are determined by the modification of maximum likelihood method for censored data. From the tests that are proper for the comparison of distribution of the duration of survival of more groups, nonparametric logrank test and parametric likelihood ratio test are mentioned. In the last section of the Bachelor thesis the theoretical findings are illustrated on simulated as well as actual data using Mathematica 9. Keywords: survival function, Kaplan-Meier estimator, logrank test, maximum likelihood method, likelihood-ratio test 1 Literatura 2 Seznam obrázků 3 Seznam tabulek 4
|
|
|
One factor models of interest rates
Jambor, Matúš ; Myška, Petr (advisor) ; Hurt, Jan (referee)
Title: One factor interest rate models Author: Matúš Jambor Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Petr Myška Abstract: In this thesis we looked closely at the models of interest rates that are applied in the area of financial mathematics and actuarial sciences. There are several models that try to describe the behavior of yield curve plausibly. In most of the cases the models stem from probability theory and coincidence. These models are also means for assessment of financial derivates whose price de- pends on the interest rates movements. The work deals with three one-factor models which are analyzed into more details in the second chapter. The last chapter is about real-data calibration. Keywords: one factor models, interest rates, maximum likelihood method 1
|
|
|
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.
|
|
|
Hypothesis Testing of interest rates models
Petrík, Daniel ; Myška, Petr (advisor) ; Hurt, Jan (referee)
V předložené práci se zabýváme problematikou stochastického modelování úro- kových sazeb. Jedním z nejobvyklejších postup· je modelovat dynamiku úroko- vých sazeb pomocí stochastické diferenciální rovnice difúze, jejímiž základními kameny jsou funkce driftu a funkce difúze. Od 70. let 20. století byla navržena celá řada model· tohoto typu, a ačkoli se tyto modely neustále zdokonalují, vyvstává přirozená otázka, zda se historicky pozorované úrokové sazby skutečně takovými difúzními rovnicemi řídily. V této práci budeme právě uvedenou hypo- tézu testovat pro několik nejběžnějších jednofaktorových model· úrokové sazby první generace. Z historických dat odhadneme obecnou momentovou metodou a metodou maximální věrohodnosti parametry jednotlivých difúzních rovnic a následně provedeme statistické testy dobré shody proložení těchto rovnic pozo- rovanými daty. 1
|
|
|
Statistical problems in Markov chains with applications in finance
Chudý, Marek ; Prokešová, Michaela (advisor) ; Pawlas, Zbyněk (referee)
Title: Statistical problems in Markov chains with applications in finance Author: Marek Chudý Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Michaela Prokešová, Ph.D. Abstract: In this work, we study estimation methods for estimating transition probabilities in Markov chains. We discuss two methods, the first one for com- plete data and the second one for aggregate data. In the second chapter, we will introduce the theory for both methods and show examples of tests of sev- eral hypothesis about transition probabilities. In the last chapter we apply both methods to real data comming from an insurance company. In the last chapter we also present the results of both methods and compare them with each other. Keywords: Markov chains, transition probabilities, maximum likelihood method, least squares method
|
|
|
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.
|
|
|
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.
|
|
|
Macroeconometric Model of Monetary Policy
Čížek, Ondřej ; Pánková, Václava (advisor) ; Kodera, Jan (referee) ; Lukáš, Ladislav (referee)
First of all, general principals of contemporary macroeconometric models are described in this dissertation together with a brief sketch of alternative approaches. Consequently, the macroeconomic model of a monetary policy is formulated in order to describe fundamental relationships between real and nominal economy. The model originated from a linear one by making some of the parameters endogenous. Despite this nonlinearity, I expressed my model in a state space form with time-varying coefficients, which can be solved by a standard Kalman filter. Using outcomes of this algorithm, likelihood function was then calculated and maximized in order to obtain estimates of the parameters. The theory of identifiability of a parametric structure is also described. Finally, the presented theory is applied on the formulated model of the euro area. In this model, the European Central Bank was assumed to behave according to the Taylor rule. The econometric estimation, however, showed that this common assumption in macroeconomic modeling is not adequate in this case. The results from econometric estimation and analysis of identifiability also indicated that the interest rate policy of the European Central Bank has only a very limited effect on real economic activity of the European Union. Both results are influential, as monetary policy in the last two decades has been modeled as interest rate policy with the Taylor rule in most macroeconometric models.
|
guest ::
