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Maximum likelihood methods; selected problems
Chlubnová, Tereza ; Hlubinka, Daniel (advisor) ; Hlávka, Zdeněk (referee)
Maximum likelihood estimation is one of statistical methods for estimating an unknown parameter. It is often used because of a simple calculation of the estimator and also for characteristics of this estimator, which the method provides under some conditions. In the thesis we prove a consistence of the estimator under conditions of regularity and uniqueness of the root of the likelihood equation. If we add other assumptions we show its asymptotic normality and we expand this result from the one-dimensional parameter to the multi-dimensional parameter. The main result of the thesis lies in exercises, in which we cannot express the maximum likelihood estimator in general, but we can show its existence, uniqueness and asymptotic normality. Moreover we demonstrate the utilization of asymptotic normality of the estimator for asymptotic hypothesis tests and confidence intervals of the parameter. Powered by TCPDF (www.tcpdf.org)
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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...
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Stochastic Differential Equations with Gaussian Noise
Janák, Josef ; Maslowski, Bohdan (advisor)
Title: Stochastic Differential Equations with Gaussian Noise Author: Josef Janák Department: Department of Probability and Mathematical Statistics Supervisor: Prof. RNDr. Bohdan Maslowski, DrSc., Department of Probability and Mathematical Statistics Abstract: Stochastic partial differential equations of second order with two un- known parameters are studied. The strongly continuous semigroup (S(t), t ≥ 0) for the hyperbolic system driven by Brownian motion is found as well as the formula for the covariance operator of the invariant measure Q (a,b) ∞ . Based on ergodicity, two suitable families of minimum contrast estimators are introduced and their strong consistency and asymptotic normality are proved. Moreover, another concept of estimation using "observation window" is studied, which leads to more families of strongly consistent estimators. Their properties and special cases are descibed as well as their asymptotic normality. The results are applied to the stochastic wave equation perturbed by Brownian noise and illustrated by several numerical simula- tions. Keywords: Stochastic hyperbolic equation, Ornstein-Uhlenbeck process, invariant measure, paramater estimation, strong consistency, asymptotic normality.
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Stochastic Differential Equations with Gaussian Noise
Janák, Josef ; Maslowski, Bohdan (advisor)
Title: Stochastic Differential Equations with Gaussian Noise Author: Josef Janák Department: Department of Probability and Mathematical Statistics Supervisor: Prof. RNDr. Bohdan Maslowski, DrSc., Department of Probability and Mathematical Statistics Abstract: Stochastic partial differential equations of second order with two un- known parameters are studied. The strongly continuous semigroup (S(t), t ≥ 0) for the hyperbolic system driven by Brownian motion is found as well as the formula for the covariance operator of the invariant measure Q (a,b) ∞ . Based on ergodicity, two suitable families of minimum contrast estimators are introduced and their strong consistency and asymptotic normality are proved. Moreover, another concept of estimation using "observation window" is studied, which leads to more families of strongly consistent estimators. Their properties and special cases are descibed as well as their asymptotic normality. The results are applied to the stochastic wave equation perturbed by Brownian noise and illustrated by several numerical simula- tions. Keywords: Stochastic hyperbolic equation, Ornstein-Uhlenbeck process, invariant measure, paramater estimation, strong consistency, asymptotic normality.
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Stochastic Differential Equations with Gaussian Noise
Janák, Josef ; Maslowski, Bohdan (advisor) ; Duncan, Tyrone E. (referee) ; Pawlas, Zbyněk (referee)
Title: Stochastic Differential Equations with Gaussian Noise Author: Josef Janák Department: Department of Probability and Mathematical Statistics Supervisor: Prof. RNDr. Bohdan Maslowski, DrSc., Department of Probability and Mathematical Statistics Abstract: Stochastic partial differential equations of second order with two un- known parameters are studied. The strongly continuous semigroup (S(t), t ≥ 0) for the hyperbolic system driven by Brownian motion is found as well as the formula for the covariance operator of the invariant measure Q (a,b) ∞ . Based on ergodicity, two suitable families of minimum contrast estimators are introduced and their strong consistency and asymptotic normality are proved. Moreover, another concept of estimation using "observation window" is studied, which leads to more families of strongly consistent estimators. Their properties and special cases are descibed as well as their asymptotic normality. The results are applied to the stochastic wave equation perturbed by Brownian noise and illustrated by several numerical simula- tions. Keywords: Stochastic hyperbolic equation, Ornstein-Uhlenbeck process, invariant measure, paramater estimation, strong consistency, asymptotic normality.
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Maximum likelihood methods; selected problems
Chlubnová, Tereza ; Hlubinka, Daniel (advisor) ; Hlávka, Zdeněk (referee)
Maximum likelihood estimation is one of statistical methods for estimating an unknown parameter. It is often used because of a simple calculation of the estimator and also for characteristics of this estimator, which the method provides under some conditions. In the thesis we prove a consistence of the estimator under conditions of regularity and uniqueness of the root of the likelihood equation. If we add other assumptions we show its asymptotic normality and we expand this result from the one-dimensional parameter to the multi-dimensional parameter. The main result of the thesis lies in exercises, in which we cannot express the maximum likelihood estimator in general, but we can show its existence, uniqueness and asymptotic normality. Moreover we demonstrate the utilization of asymptotic normality of the estimator for asymptotic hypothesis tests and confidence intervals of the parameter. Powered by TCPDF (www.tcpdf.org)
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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...
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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...
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