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Estimators of variance function in nonparametric regression
Hyklová, Bronislava ; Antoch, Jaromír (referee) ; Hušková, Marie (advisor)
The thesis studies variance function estimation in nonparametric regression model. It focuses on local polynomial estimators particularly. Exact expressions of conditional variance function estimator bias and covariance are derived and important asymptotical aproximations of these characteristics are also provided. Further the EBBS method for bandwidth selection and Dette's homoscedasticity test are described. Results of Prague Klementinum data processing are presented at the end of the thesis.
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Tests based on U-statistics
Madurkayová, Barbora ; Jurečková, Jana (referee) ; Hušková, Marie (advisor)
There are many test statistics that are based on U-statistics. This thesis deals with tests based on U-statistics for the general two-sample problem. After describing two-sample tests based on U-statistics from a general viewpoint, a presentation of some particular test statistics follows. All considered test statistics are described in a connection to the theory of U-statistics, with emphasis on their asymptotic properties. Special concern is given to tests based on a difference between two one-sample U-statistics and to tests based on empirical characteristic functions. Test statistics, based on a difference of two empirical characteristic functions, have a form of a V -statistic. A related U-statistic is derived and its properties are studied. An example of applying the bootstrap method to these test statistics is included.
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Estimates based on the rank tests
Setíkovská, Jitka ; Hušková, Marie (referee) ; Jurečková, Jana (advisor)
The thesis deals with R-estimators, estimators based on ranks. They were originally proposed by Hodges and Lehmann [7] as inversions of the rank tests. Not only the definition used by Hodges and Lehmann, but also the one used in later literature is formulated. Basic characteristics of some rank statistics are described and the explicit forms of the corresponding R-estimators and confidence intervals are derived. Their basic properties as unbiasedness, translation equivariance, efficiency, robustness are studied. The behavior of R-estimators is then illustrated on simulated data and on several examples.
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