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Chain-ladder method as maximum likelihood estimator in Poisson model
Wagner, Vojtěch ; Kříž, Pavel (advisor) ; Pešta, Michal (referee)
First, the distribution-free chain-ladder is introduced. Then, the Poisson model is in- troduced. It is proven that the total reserves for one accident year given by the maximum likelihood estimation applied to the Poisson model lead to the identical reserves as the reserves derived from the distribution-free chain-ladder used in the Poisson model. Later, inadequacies of the Poisson model are discussed. Hessian matrices of the log-likelihood evaluated at the Poisson estimators are analyzed. The question whether the inverse of the Fisher information matrix approximates the real covariance matrix of the Poisson esti- mators is explored. Comparing the variance of the total reserves derived from the inverse of the Fisher information and the real covariance matrix leads to negative conclusion, that the former does not approximate the latter well. 1
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Communication complexity
Wagner, Vojtěch ; Krajíček, Jan (advisor) ; Koucký, Michal (referee)
Title: Communication Complexity Author: Vojtěch Wagner Department: Department of Algebra Supervisor: prof. RNDr. Jan Krajíček, DrSc. Abstract: This work deals with communication complexity, which considers mo- del of two (or more) parties, each holding its own binary input (let's say x and y). Each of players has information only about his own input. Their common goal is to compute value of some function f(x, y) of these inputs. Communicati- on complexity measures amount of information communicated between players in order to compute f(x, y). This work especially concerns two main models - deterministic, in which all decision made by players is deterministic and they compute the right value in all cases and probabilistic model which allows ran- domized fashion and the goal of the players is to compute the right value with high enough probability. We present some basic concepts and methods to lower bound communication complexity of functions, all ilustrated on some examples of basic functions. In the end we present some complex and practically relevant examples, on which presented methods are demonstrated. Keywords: communication complexity, deterministic communication model, ran- domized model, lower bounds on communication complexity. 1
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Logika a kryptografie
Wagner, Vojtěch ; Krajíček, Jan (advisor) ; Thapen, Neil (referee)
Title: Logic and cryptography Author: Bc.Vojtěch Wagner Department: Department of Algebra Supervisor: prof. RNDr. Jan Krajíček, DrSc. Abstract: This work is devoted to a study of a formal method of formalization of cryptographic constructions. It is based on defining a multi-sorted formal logic theory T composed of strings, integers and objects of sort k - k-ary functions. We allow some operations on them, formulate axioms, terms and formulas. We also have a special type of integers called the counting integers. It denotes the number of x from a given interval satisfying formula ϕ(x). It allows us to talk about probabilities and use terms of probability theory. The work first describes this theory and then it brings a formalization of the Goldreich-Levin theorem. The goal of this work is to adapt all needed cryptographic terms into the language of T and then prove the theorem using objects, rules and axioms of T. Presented definitions and principles are ilustrated on examples. The purpose of this work is to show that such theory is sufficiently strong to prove such cryptographic constructions and verify its correctness and security. Keywords: cryptography, protocol verifying, Soundness theorem, formal logic theory, the Goldreich-Levin theorem 1
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Communication complexity
Wagner, Vojtěch ; Krajíček, Jan (advisor) ; Koucký, Michal (referee)
Title: Communication Complexity Author: Vojtěch Wagner Department: Department of Algebra Supervisor: prof. RNDr. Jan Krajíček, DrSc. Abstract: This work deals with communication complexity, which considers mo- del of two (or more) parties, each holding its own binary input (let's say x and y). Each of players has information only about his own input. Their common goal is to compute value of some function f(x, y) of these inputs. Communicati- on complexity measures amount of information communicated between players in order to compute f(x, y). This work especially concerns two main models - deterministic, in which all decision made by players is deterministic and they compute the right value in all cases and probabilistic model which allows ran- domized fashion and the goal of the players is to compute the right value with high enough probability. We present some basic concepts and methods to lower bound communication complexity of functions, all ilustrated on some examples of basic functions. In the end we present some complex and practically relevant examples, on which presented methods are demonstrated. Keywords: communication complexity, deterministic communication model, ran- domized model, lower bounds on communication complexity. 1
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