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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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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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Structural Equation Models with Application in Social Sciences
Veselý, Václav ; Pešta, Michal (advisor) ; Maciak, Matúš (referee)
We investigate possible usage of Errors-in-Variables estimator (EIV), when esti- mating structural equations models (SEM). Structural equations modelling pro- vides framework for analysing complex relations among set of random variables where for example the response variable in one equation plays role of the predic- tor in another equation. First an overview of SEM and some common covariance based estimators is provided. Special case of linear regression model is investi- gated, showing that the covariance based estimators yield the same results as ordinary least squares. A compact review of EIV models follows, Errors-in-Variables models are re- gression models where not only response but also predictors are assumed to be measured with an error. Main contribution of this paper then lies in defining modifications of the EIV estimator to fit in the SEM framework. General opti- mization problem to estimate the parameters of structural equations model with errors-in-variables si postulated. Several modifications of two stage least squares are also proposed for future research. Equation-wise Errors-in-Variables estimator is proposed to estimate the coeffi- cients of structural equations model. The coefficients of every structural equation are estimated separately using EIV estimator. Some theoretical conditions...
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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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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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