|
|
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.
|
|
|
Operational risk loss distributions
Krajňák, Tomáš ; Mazurová, Lucie (advisor) ; Pešta, Michal (referee)
Operational risk in recent years has become an important part of banks, insurance companies and financial institutions. The proposed work deals with the distributions that best fit the loss severity from the operational risk and also describe their basic properties. Specifically, deals with the g-h distribution, its properties, moments, parameter estimations and tail behavior. There is also another method for high threshold estimation described in this text, the POT (Peaks over threshold). In conclusion, there is the procedure for estimating quantiles of g-h distribution by POT method presented including simulation example in which there are quantile values estimated using the POT method compared to the g-h distribution quantiles.
|
|
|
Credibility approach to claims reserves calculation
Dzugas, Erik ; Mazurová, Lucie (advisor) ; Pešta, Michal (referee)
In this work we summarize the various techniques of claims reserves evaluating which consist in estimate of the future uncertain and hardly antici- pated loss development. It appears that the methods which are based on some credibility formula bring in the mean squared error sense the most accurate results. We consider this in the text derived conclusion very relevant and con- tributing, therefore we illustrate and present it on the numerical example. The calculations are introduced in the attached charts that build the important sup- plement of the text. The topic of this work follows up the content of Nonlife Insurance and Risk Theory lectures, therefore this text can be useful also for the students of the Faculty of Mathematics and Physics to extend their knowledge. 1
|
|
|
Modern Asymptotic Perspectives on Errors-in-variables Modeling
Pešta, Michal ; Antoch, Jaromír (advisor) ; Lachout, Petr (referee) ; Zwanzig, Silvelyn (referee)
A linear regression model, where covariates and a response are subject to errors, is considered in this thesis. For so-called errors-in-variables (EIV) model, suitable error structures are proposed, various unknown parameter estimation techniques are performed, and recent algebraic and statistical results are summarized. An extension of the total least squares (TLS) estimate in the EIV model-the EIV estimate-is invented. Its invariant (with respect to scale) and equivariant (with respect to the covariates' rotation, to the change of covariates direction, and to the interchange of covariates) properties are derived. Moreover, it is shown that the EIV estimate coincides with any unitarily invariant penalizing solution to the EIV problem. It is demonstrated that the asymptotic normality of the EIV estimate is computationally useless for a construction of confidence intervals or hypothesis testing. A proper bootstrap procedure is constructed to overcome such an issue. The validity of the bootstrap technique is proved. A simulation study and a real data example assure of its appropriateness. Strong and uniformly strong mixing errors are taken into account instead of the independent ones. For such a case, the strong consistency and the asymptotic normality of the EIV estimate are shown. Despite of that, their...
|
|
|
Modern Asymptotic Perspectives on Errors-in-variables Modeling
Pešta, Michal
Charles University in Prague Faculty of Mathematics and Physics ABSTRACT OF DOCTORAL THESIS Michal Pešta MODERN ASYMPTOTIC PERSPECTIVES ON ERRORS-IN-VARIABLES MODELING A linear regression model, where covariates and a response are subject to errors, is considered in this thesis. For so-called errors-in-variables (EIV) model, suitable error structures are proposed, various unknown parameter estimation techniques are performed, and recent algebraic and statistical results are summarized. An extension of the total least squares (TLS) estimate in the EIV model-the EIV estimate-is in- vented. Its invariant (with respect to scale) and equivariant (with respect to the covariates' rotation, to the change of covariates direction, and to the interchange of covariates) properties are derived. Moreover, it is shown that the EIV estimate coincides with any unitarily invariant penalizing solution to the EIV problem. It is demonstrated that the asymptotic normality of the EIV estimate is computationally useless for a construction of confidence intervals or hypothesis testing. A proper bootstrap procedure is constructed to overcome such an issue. The validity of the bootstrap technique is proved. A simulation study and a real data example assure of its appropriateness. Strong and uniformly strong mixing errors are taken...
|
|
| |
|
| |
|
| |
|
| |
|
|
Isotonic Regression in Sobolev Spaces
Pešta, Michal ; Dostál, Petr (referee) ; Hlávka, Zdeněk (advisor)
We propose a class of nonparametric estimators for the regression models based on least squares over the sets of sufficiently smooth functions. Least squares permit the imposition of additional constraint-isotonia-on nonparametric regression estimation and testing of this constraint. The estimation takes place over the balls of functions which are elements of a suitable Sobolev space-special types of Hilbert spaces that facilitate calculation of the least squares projection. The Hilbertness is allowing us to take projections and hence to decompose spaces into mutually orthogonal complements. Then we transform the problem of searching for the best fitting function in an infinite dimensional space into a finite dimensional optimization problem. We prove that the balls of functions in Sobolev space are bounded and have bounded higher order derivatives. It permits us to estimate over rich set of functions with sufficiently low metric entropy and apply Laws of Large Numbers and Central Limit Theorems.
|
guest ::
RSS feed.