
Parameter estimation of gamma distribution
Zahrádková, Petra ; Kulich, Michal (advisor) ; Hlávka, Zdeněk (referee)
It is wellknown that maximum likelihood (ML) estimators of the two parame ters in a Gamma distribution do not have closed forms. The Gamma distribution is a special case of a generalized Gamma distribution. Two of the three likeli hood equations of the generalized Gamma distribution can be used as estimating equations for the Gamma distribution, based on which simple closedform estima tors for the two Gamma parameters are available. Intuitively, performance of the new estimators based on likelihood equations should be close to the ML estima tors. The study consolidates this conjecture by establishing the asymptotic beha viours of the new estimators. In addition, the closedforms enable biascorrections to these estimators. 1


Sums of exponential random variables
Michl, Marek ; Seidler, Jan (advisor) ; Maslowski, Bohdan (referee)
Sums of exponential random variables are often found in applied mathematics. Their densities are known and are well documented in engineering articles. However, these articles usually lack detailed deductions. The purpose of this thesis is to give rigorous deductions of explicit formulas for densities of sums of independent exponential random variables, which are known. The thesis covers several cases depending on whether the variables have the same distribution or not. Furthermore, the thesis gives a summary of basic characteristics of exponential distribution and the relation between sums of identically distributed exponential random variables and a random variable with gamma distribution. Based on this relation the density of the sum of gamma random variables with the same intestity is given. Powered by TCPDF (www.tcpdf.org)


Highorder stochastic dominance
Mikulka, Jakub ; Kopa, Miloš (advisor) ; Branda, Martin (referee)
The thesis deals with highorder stochastic dominance of random variables and portfolios. The summary of findings about highorder stochastic dominance and portfolio efficiency is presented. As a main part of the thesis it is proven that under assumption of both normal and gamma distribution the infiniteorder stochastic dominance is equivalent to the secondorder stochastic dominance. The necessary and sufficient condition for the infiniteorder stochastic dominance portfolio efficiency is derived under the assumption of normality. The condition is used in the empirical part of the thesis where parametrical approach to the portfolio efficiency is compared to the nonparametric scenario approach. The derived necessary and sufficient condition is based on the assumption of normality; therefore we use two sets of data, one with fulfilled assumption of normality and the other for which the assumption of normality was unambigously rejected. Consequently, the influence of fulfillment of the normality assumption on the results of the necessary and sufficient condition for portfolio efficiency is estimated.
