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Chebyshev inequality and some its modifications
Drabinová, Adéla ; Anděl, Jiří (advisor) ; Nagy, Stanislav (referee)
In the presented thesis we describe some improvements of Chebyshev inequa- lity. In the first chapter we introduce inequalities for random variables with uni- modal distributions. We prove Gauss and Camp-Meidell inequality and we deduce Vysochanskii-Petunin inequality. We describe inequalities for variables with mode 0 and with unspecified mode. In the second chapter we consider constants C(r), for which the approximations are the best. We are interested in finding optimal parameter r or its approximation. In the third chapter we state inequalities from the first chapter for specific distributions, calculation of their constants, appli- cations and graphic presentations of the results. 1
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Nonlinearity in time series models
Kalibán, František ; Anděl, Jiří (advisor) ; Zvára, Karel (referee)
The thesis concentrates on property of linearity in time series models, its definitions and possibilities of testing. Presented tests focus mainly on the time domain; these are based on various statistical methods such as regression, neural networks and random fields. Their implementation in R software is described. Advantages and disadvantages for tests, which are implemented in more than one package, are discussed. Second topic of the thesis is additivity in nonlinear models. The definition is introduced as well as tests developed for testing its presence. Several test (both linearity and additivity) have been implemented in R for purposes of simulations. The last chapter deals with application of tests to real data. 1
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Statistical analysis of historical temperature series
Gergelits, Václav ; Antoch, Jaromír (advisor) ; Anděl, Jiří (referee)
Title: Statistical analysis of historical temperature series Author: Václav Gergelits Department: Department of Probability and Mathematical Statistics Supervisor: prof. RNDr. Jaromír Antoch CSc. Supervisor's e-mail address: antoch@karlin.mff.cuni.cz Abstract: In the present work we deal with the statistical analysis of time-series of a mean-temperature obtained from seven European cities from the Europe Union project "IMPROVE". Properties of the time series are analyzed by means of descriptive statistics, being assessing their homoscedasticity, autocorrelation and normality. We report the ways in which the data has been adjusted, including consideration of the impact of the urban heat island and we discuss the availability of additional data. The theoretical part presents a theory of change point detection for a one change model as well as more than one change model taking an autocorrelation into account. In the practical part we analyze the data using change point detection method. The significant increase was not detected for time series of Cadiz and Uppsala. The significant increase was rather detected for the rest of the time series. The increase of temperature could be in a relation to the adjustment for the urban heat island. Keywords: change point detection, temperature time series 1
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Some functions of ARMA processes
Štufka, Miroslav ; Anděl, Jiří (advisor) ; Prášková, Zuzana (referee)
This study provides a comprehensive overview of changes in the autoregressive-moving- average model (ARMA) when applied to various functions. First, the necessary and sufficient condition for a weakly stationary stochastic process described by ARMA is given. Next, some particular transformations of ARMA processes are presented: first, non- correlated and generic sums of ARMA processes; next, products of independent and dependent Gaussian ARMA processes; and finally, time aggregations, namely, systematic sampling and temporal aggregations. Tables are included to clearly summarize special cases of particular transformations. Some of these cases are then demonstrated through concrete examples. In addition to theoretical results, extensive numerical simulation in statistical software R is also given, which systematically covers the obtained results.
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Birthday problem
Drápal, Lukáš ; Anděl, Jiří (advisor) ; Dostál, Petr (referee)
In the presented work we discuss the birthday problem with unlike probabilities. First, we introduce the concept of majorization of vectors, Schur convexity of functions and Bell polynomials. Using these concepts we show the results from papers [6] and [8]. We also discuss the paper [7] and we point out its error. Then we present a program in language R that is simulating the problem. We use this program to calculate the probability for the true birthday problem in the Czech Republic and the effect of leap years. Finally, we show some applications of the birthday problem, especially the true surname problem in Japan [8].
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Weighted Data Depth and Depth Based Discrimination
Vencálek, Ondřej ; Hlubinka, Daniel (advisor) ; Anděl, Jiří (referee) ; Malý, Marek (referee)
The concept of data depth provides a powerful nonparametric tool for multivariate data analysis. We propose a generalization of the well-known halfspace depth called weighted data depth. The weighted data depth is not affine invariant in general, but it has some useful properties as possible nonconvex central areas. We further discuss application of data depth methodology to solve discrimination problem. Several classifiers based on data depth are reviewed and one new classifier is proposed. The new classifier is a modification of k-nearest- neighbour classifier. Classifiers are compared in a short simulation study. Advantage gained from use of the weighted data depth for discrimination purposes is shown.
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Normality tests
Kotlorz, Lukáš ; Anděl, Jiří (advisor) ; Sabolová, Radka (referee)
The aim of this thesis focused on testing normality is to describe both statistical tests and graphical methods. The first part is devoted to graphical methods used to testing normality (particularly Histogram, Boxplot and Q-Q Plot). The tests used for testing the conformity of random sample distribution with normal distribution, e.g., Shapiro-Wilk, Kolmogorov-Smirnov, Lilliefors, Anderson-Darling, Chi-squared, are described in the second part. The test statistics, the critical region and alternatively the link for tabulated critical values are listed for each test. The simulations, whether the random sample comes from normal distribution, are described in the third part. The samples from di erent distributions were generated by Program R.
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Errors-in-variables models
Fürjesová, Ida ; Pešta, Michal (advisor) ; Anděl, Jiří (referee)
This thesis analyzes an errors-in-variables model. It compares parameter estimation methods least squares and total least squares. The main difference between these methods lies in approach to the measurements errors. The first part of the bachelor thesis focuses on theoretical aspect of methods. It defines basic terms and shows differences in the methods graphically. Thesis also demonstrates algebraic solutions of the estimation methods. The theoretical part ends up with statistical properties of the estimating techniques. The thesis compares methods least squares and total least squares according to the size of mean square error by simulation study.
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Success runs in series of Bernoulli trials
Mach, Tibor ; Anděl, Jiří (advisor) ; Dvořák, Marek (referee)
This work is focused on selected probability characteristics of runs in a sequence of Bernoulli trials and on some randomness tests based on these runs. Based on Markov chains, an explicit formula is derived for the probability that the first success run of a lenght $k$ in a sequence of independent Bernoulli trials occurs in the $n$-th trial and other formulas for this probability are mentioned. Furthermore, approximations of the exact value of this probability (particularly the Feller approximation), bounds of these approximations, and their numeric relations are examined. Lastly, a test of randomness based on the lenght of the longest run in a sequence of $n$ Bernoulli trials and a test based on the total amount of runs are derived.
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Tests in multinomial distribution
Holý, Vladimír ; Anděl, Jiří (advisor) ; Antoch, Jaromír (referee)
In this paper there are at first described classical goodness-of-fit tests - the Pear- son's χ2 test and the log likehood ratio test. The more modern method of testing is the family of statistics based on power divergence which is generalisation of classical statistics. Another type of generalisation is the family of disparity statis- tics which includes beside the family of power divergence also the families BWHD and BWCS. It is demonstrated that all these test statistics have an asymptotic χ2 distribution. In the program R the exact level and exact power can be calculated for individual tests. Hereafter, moments of test statistics can be derived. On the basis of these comparisons there will be shown which test statistics are the most suitable for the goodness-of-fit tests. 1
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