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Estimation of the pair correlation function of a point process
Vondráček, Jakub ; Dvořák, Jiří (advisor) ; Beneš, Viktor (referee)
This thesis deals with kernel estimation of the pair correlation function of a stationary and isotropic point process. Firstly, the basics of the theory of point processes are built up. Then, the derivation of formulas for expectation and variance of a kernel estimator is provided. Also, an extension of a simple Poisson approximation of variance to the case of an estimator with more complicated edge correction compared to what is usually used in the literature is given. These formulas depend on a parameter called bandwidth. The recommendations for selecting the bandwidth that can be found in the literature are summarised and simulation experiments are performed to assess the correctness of the derived formulas. These experiments also prove that a variance approximation obtained by ignoring so called "higher order correlations" is unjustified. Lastly, bandwidth selection and the advantages and disadvantages of several approaches for bandwidth selection are discussed. 1
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Estimator averaging
Plotnikova, Valeriya ; Dvořák, Jiří (advisor) ; Čoupek, Petr (referee)
We investigate general procedure to combine several estimators of the same real parameter in the parametric model. Considering weighted average of initial estimator, where weights are constrained to the sum of one, we obtain the final combined estimator. In that framework, the optimal vector of weights minimises the mean square error, providing better estimator of parameter. Combined es- timator will be computed as the product of vector of weights and the vector of initial estimators of the parameter. This method is frequently used in numerous problems of modern statistics like forecast averaging for time series.
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Optimality of sample variance
Gleta, Filip ; Kulich, Michal (advisor) ; Anděl, Jiří (referee)
It is widely known that the most common estimators of the variance and the standard deviation based on i.i.d. data are not optimal with respect to the mean squared error. The aim of this thesis is to study and summarize the various approaches to seeking an improved estimator, which stem mainly from the innovative ideas presented by Stein (1964). Taken into consideration is the point estimator of the variance and the standard deviation. Each of the improved estimators include, in addition to their construction, a discussion regarding admissibility with respect to the MSE. Subsequently, using simple simulations for various distributions, it is examined whether obtained improvements lead to better results in practice. Powered by TCPDF (www.tcpdf.org)
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Statistical hypothesis testing using sports data
Černý, Jakub ; Malá, Ivana (advisor) ; Čabla, Adam (referee)
This bachelor thesis is based on empirical study of betting odds of six world-wide betting offices. Using statistical hypothesis testing it tests similarity among offices predictions of five best football leagues and whether the predictions correspond with final results of these games. It has been demonstrated that betting offices predict matches identically but predictions does not correspond with final results. In the second part of this paper, the betting offices are experimentally treated as bettors and their return on investment is monitored. From the comparison of returns and mean squared error of predictions, betting offices Bet365 and BetVictor can be considered as those with the best predictions. Analysis of Fortel servis betting diary has shown that using betting advisory service can lead to long-term profits. It is a proof that betting offices can be beaten over a long time which was also one of the objectives of this paper.
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Economics of Biased Estimation
Drvoštěp, Tomáš ; Špecián, Petr (advisor) ; Tříska, Dušan (referee)
This thesis investigates optimality of heuristic forecasting. According to Goldstein a Gigerenzer (2009), heuristics can be viewed as predictive models, whose simplicity is exploiting the bias-variance trade-off. Economic agents learning in the context of rational expectations (Marcet a Sargent 1989) employ, on the contrary, complex models of the whole economy. Both of these approaches can be perceived as an optimal response complexity of the prediction task and availability of observations. This work introduces a straightforward extension to the standard model of decision making under uncertainty, where agents utility depends on accuracy of their predictions and where model complexity is moderated by regularization parameter. Results of Monte Carlo simulations reveal that in complicated environments, where few observations are at disposal, it is beneficial to construct simple models resembling heuristics. Unbiased models are preferred in more convenient conditions.
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