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Admissibility and Inadmissibility of an Estimate
Vagner, Marcel ; Maciak, Matúš (advisor) ; Jurečková, Jana (referee)
The quality of a parameter estimate is usually assessed using the mean squared error (MSE). For one dimensional parameter, the estimate constructed using the least squares method is the best. However, for a vector parameter with more than two dimensions this estimator becomes inadmissible. There is always some different estimator which domi- nates the least squares estimate regardless of the parameter value. This phenomenon is well known as the Stein Paradox. The aim of this bachelor thesis is to describe admissi- bility and inadmissibility of an estimator, define the James-Stein estimator and perform a simulation study to compare different estimators. 1
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Tests of statistical hypotheses in measurement error models
Navrátil, Radim ; Jurečková, Jana (advisor) ; Hušková, Marie (referee) ; Kalina, Jan (referee)
The behavior of rank procedures in measurement error models was studied - if tests and estimates stay valid and applicable when there are some measurement errors involved and if not how to modify these procedures to be able to do some statistical inference. A new rank test for the slope parameter in regression model based on minimum distance esti- mator and an aligned rank test for an intercept were proposed. The (asymptotic) bias of R-estimator in measurement error model was also investigated. Besides measurement errors the problem of heteroscedastic model errors was considered - regression rank score tests of heteroscedasticity with nuisance regression and tests of regression with nuisance heterosce- dasticity were proposed. Finally, in location model tests and estimates of shift parameter for various measurement errors were studied. All the results were derived theoretically and then demonstrated numerically with examples or simulations.
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Statistické odhady a chvosty jejich rozdělení pravděpodobností
Veverková, Jana ; Jurečková, Jana (advisor) ; Antoch, Jaromír (referee)
Master Thesis Statistical estimators and their tail behavior provides description of two type of characteristics of robustness of estimators - tail behavior and break- down point. Description is made for translation equivariant estimators in general and also for some concrete type of estimators, sample mean, sample median, trimmed mean, Huber estimator and Hodges Lehmann estimator. Tail behavior of these estimator is illustrated for random sample coming from t-distribution with 1 to 5 degrees of freedom. Ilustration is based on simulations made in Mathematica. 1
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Statistical inference based on saddlepoint approximations
Sabolová, Radka ; Jurečková, Jana (advisor) ; Hlávka, Zdeněk (referee) ; Picek, Jan (referee)
Title: Statistical inference based on saddlepoint approximations Author: Radka Sabolová Abstract: The saddlepoint techniques for M-estimators have proved to be very accurate and robust even for small sample sizes. Based on these results, saddle- point approximations of density of regression quantile and saddlepoint tests on the value of regression quantile were derived, both in parametric and nonpara- metric setup. Among these, a test on the value of regression quantile based on the asymptotic distribution of averaged regression quantiles was also proposed and all these tests were compared in a numerical study to the classical tests. Finally, special case of Kullback-Leibler divergence in exponential family was studied and saddlepoint approximations of the density of maximum likelihood estimator and sufficient statistic were also derived using this divergence. 1
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Sampling methods in forestry
Hanek, Petr ; Pawlas, Zbyněk (advisor) ; Jurečková, Jana (referee)
This diploma thesis is devoted to the sampling strategies in forestry. It describes their theoretical aspects and their applications on a real landscape. The sampling methods in forestry are of particular importance in forest inven- tory. The aim of sampling methods is to estimate population characteristics based on the knowledge of sample. Two basic approaches can be distinguished according to the size of population, we speak about discrete or continuous population. Several types of sampling designs and corresponding estimators of target values are described for both approaches. Besides estimates of po- pulation total or average, we mention the formulas for computing variance of these estimates and the methods for their estimation for different sampling designs. The thesis also contains the comparison of studied methods based on computer simulations.
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Stochastical inference in the model of extreme events
Dienstbier, Jan ; Picek, Jan (advisor) ; Jurečková, Jana (referee) ; Jarušková, Daniela (referee)
Title: Stochastical inference in the model of extreme events Author: Jan Dienstbier Department/Institute: Department of probability and mathematical statistics Supervisor of the doctoral thesis: Doc. RNDr. Jan Picek, CSc. Abstract: The thesis deals with extremal aspects of linear models. We provide a brief explanation of extreme value theory. The attention is then turned to linear models Yn×1 = Xn×pβp×1 + En×1 with the errors Ei ∼ F, i = 1, . . . , n fulfilling the do- main of attraction condition. We examine the properties of the regression quantiles of Koenker and Basset (1978) under this setting we develop theory dealing with extremal characteristics of linear models. Our methods are based on an approximation of the regression quantile process for α ∈ [0, 1] expanding older results of Gutenbrunner et al. (1993). Our result holds in [α∗ n, 1 − α∗ n] with a better rate of α∗ n → 0 than the other approximations described previously in the literature. Consecutively we provide an ap- proximation of the tails of regression quantile. The approximations of the tails enable to develop theory of the smooth functionals, which are used to establish a new class of estimates of extreme value index. We prove T(F−1 n (1 − knt/n)) is consistent and asymp- totically normal estimate of extreme for any T member of the class....
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Effect of covariate measurement error on estimates and tests in regression models
Otčenášová, Eliška ; Kulich, Michal (advisor) ; Jurečková, Jana (referee)
The thesis concerns with e ect of covariate measurement error on the least squares estimators and tests of importance of parameters in regression models. It refers to unsatis ed assumptions of linear model when using measurement error covariates and resulting e ect on estimates and tests in regression models. It focuses mainly on investigation of consistence of estimates of linear and quadratic coeficient in additive and multiplicative model with one covariate with homoscedastic and heteroscedastic measurement error. In the nal chapter teoretical results are grounded by simulation study.
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The uniformly most powerful test. The uniformly most powerful unbiased test
Sečkárová, Vladimíra ; Juríček, Jozef (advisor) ; Jurečková, Jana (referee)
Nazov prace: Stejuomrrne ncjsilnejsi test. Stejnomerne nojsilnejsi nestranny test Autor: Vladimira Seckarova Katcdra: Katedra pravdepodobnosti a matematickej statistiky Veduci bakalarskoj prace: Mgr. .Jozef Juricek e-mail voduceho: jurijlam@artax.karlin.mff.cuni.cz Al>strakt: Tato prtica sa zaobera prol)letnatikou testovania hypotez. konkretne existenciou rovnonierne najsilnejsieho a rovnomerne najsilnej.sicho nestrauueho testu. Prva kapitola olisahuje zakladno pojiny testovania hypote/. V druhej ka- pitole jo zayodcny pojein rovnomerne najsilnejsieho testu ako i jeho odvodenie v roznyeh obeenych ]>ripadoch i pro pa.rametre normalneho roxdelenia. Trctia. ka- pitola sa zaobera rovnomerne najsilnejsini ncstrannyin testom a jeho odvodonim obocnc a aj ])re parn.met.ro normalneho ro/,delenia. KlYieove .slov;i: testovanie hypotez, najsilnejsi test, rovnomerne naj.silnejsi test, rovnomerne najsilnejsi nc.stranny test Title: Uniformly most powerful test. Uniformly most powerful unbiased test Author: Vladimira Seckarova Department: Department of Probability and Mathematical Statistics Supervisor: Mgr. .lozef .luricek Supervisor's e-mail address: jurijlam@artax.karlin.mff.cuni.cz Abstract: In this work we study problems of hypotheses testing, particularly exis- tence of the uniformly most powerful and the uniformly...
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