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Modelování extrémních hodnot
Shykhmanter, Dmytro ; Malá, Ivana (advisor) ; Luknár, Ivan (referee)
Modeling of extreme events is a challenging statistical task. Firstly, there is always a limit number of observations and secondly therefore no experience to back test the result. One way of estimating higher quantiles is to fit one of theoretical distributions to the data and extrapolate to the tail. The shortcoming of this approach is that the estimate of the tail is based on the observations in the center of distribution. Alternative approach to this problem is based on idea to split the data into two sub-populations and model body of the distribution separately from the tail. This methodology is applied to non-life insurance losses, where extremes are particularly important for risk management. Never the less, even this approach is not a conclusive solution of heavy tail modeling. In either case, estimated 99.5% percentiles have such high standard errors, that the their reliability is very low. On the other hand this approach is theoretically valid and deserves to be considered as one of the possible methods of extreme value analysis.
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Frequentist and Bayesian inference
Shykhmanter, Dmytro ; Vilikus, Ondřej (advisor) ; Hebák, Petr (referee)
The thesis provides both theoretical and practical comparison of frequentist and Bayesian methods of statistical inference. Comparing of these two concepts begins with describing the philosophy of probability theory. Also is introduced the problem of determinism as well as three main probability interpretations. Statistical inference is a process of making general conclusions based on a given evidence. The frequentist statistics uses the observed data as an only evidence for its conclusions, while the Bayesian one is based on an idea that the subjective degree of belief can be also used for these purposes. Why should one disregard to his experience, knowledge or even intuition? Often happens that results of statistical data analysis are useless in sense that they come out not as it is expected. This situation is illustrated when there are a number of ski resorts which are graded on five star scale. If we look to the top ten, we will find that some of those should not belong there, though the data says they do. Generally the top positions are occupied by the objects with fewer reviews, while those with more reviews get lower average score. Bayesian data analysis methods enable to eliminate this kind of problem. Based on a prior information about the whole data set, every ski resort would get a fair score and as the result, the model would better represent the quality of the each resort based on the respondents' reviews.
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