The thesis describes the theory of capital asset pricing model (CAPM) and the issue of robust estimates. Robust methods are an effective tool to achieve better estimation relative to the classical least squares method when there is a fai- lure to assume a normal distribution of errors or in the presence of outlying obser- vations in the data. Theory of M-estimates, which is then applied in the practical part of the thesis to the multidimensional CAPM model is treated in detail. The- ory of R- and L-estimates is explained in less detail. A simulation study compares simultaneous estimates in multivariate model and estimates designed individually when applied to the model assuming the mutual independence of equations. 1

The aim of this thesis is to analyze the properties of robust estimates and to compare these estimates with regard to the properties between them. The analysis of properties depends on the type of a variable (continuous or discrete), its probability distribution, the range of random sample and the proportion of outliers in random sample. Comparing the properties in different situations will give "guidance" to determine which of the estimates is preferable in specific situations, and which of them should be rather avoided. An adjusted bootstrap method is used to obtain the estimates of properties estimates. The thesis is devided into two parts. In the first part, the parameter estimates, type and design of robust estimators and the bootstrap method are monitored. In the second practical part we determine the suitability of bootstrap to obtain estimates of the properties of robust estimators, followed by obtaining estimates of the properties of the estimates and compare them. At the conclusion of the practical part we observe and compare the values of bootstrap confidence intervals on real data on household income. The results of this thesis shows us, that the bootstrap method does not provide good estimates of the properties of robust estimators in all cases. The results also bring us to the conclusion that from a certain extent of random sample regardless of the number of outliers, you can choose from a robust estimate only on the basis of its value, properties of robust estimates are very similar. Contemplated robust estimates of variability are not suitable estimates in most cases.

Neste trabalho estendemos os modelos mistos semiparamétricos propostos por Zhang et al. (1998) para uma classe mais geral de modelos, a qual denominamos modelos mistos aditivos semiparamétricos com erros de contornos elípticos. Com essa nova abordagem, flexibilizamos a curtose da distribuição dos erros possibilitando a escolha de distribuições com caudas mais leves ou mais pesadas do que as caudas da distribuição normal padrão. Funções de verossimilhança penalizadas são aplicadas para a obtenção das estimativas de máxima verossimilhança com os respectivos erros padrão aproximados. Essas estimativas, sob erros de caudas pesadas, são robustas no sentido da distância de Mahalanobis contra observações aberrantes. Curvaturas de influência local são obtidas segundo alguns esquemas de perturbação e gráficos de diagnóstico são propostos. Exemplos ilustrativos são apresentados em que ajustes sob erros normais são comparados, através das metodologias de sensibilidade desenvolvidas no trabalho, com ajustes sob erros de contornos elípticos. / In this work we extend the models proposed by Zhang et al. (1998) to a more general class of models, know as semiparametric additive mixed models with elliptical errors in order to allow distributions with heavier or lighter tails than the normal ones. Penalized likelihood equations are applied to derive the maximum likelihood estimates which appear to be robust against outlying observations in the sense of the Mahalanobis distance. In order to study the sensitivity of the penalized estimates under some usual perturbation schemes in the model or data, the local influence curvatures are derived and some diagnostic graphics are proposed. Motivating examples preliminary analyzed under normal errors are reanalyzed under some appropriate elliptical errors. The local influence approach is used to compare the sensitivity of the model estimates.

Neste trabalho estendemos os modelos mistos semiparamétricos propostos por Zhang et al. (1998) para uma classe mais geral de modelos, a qual denominamos modelos mistos aditivos semiparamétricos com erros de contornos elípticos. Com essa nova abordagem, flexibilizamos a curtose da distribuição dos erros possibilitando a escolha de distribuições com caudas mais leves ou mais pesadas do que as caudas da distribuição normal padrão. Funções de verossimilhança penalizadas são aplicadas para a obtenção das estimativas de máxima verossimilhança com os respectivos erros padrão aproximados. Essas estimativas, sob erros de caudas pesadas, são robustas no sentido da distância de Mahalanobis contra observações aberrantes. Curvaturas de influência local são obtidas segundo alguns esquemas de perturbação e gráficos de diagnóstico são propostos. Exemplos ilustrativos são apresentados em que ajustes sob erros normais são comparados, através das metodologias de sensibilidade desenvolvidas no trabalho, com ajustes sob erros de contornos elípticos. / In this work we extend the models proposed by Zhang et al. (1998) to a more general class of models, know as semiparametric additive mixed models with elliptical errors in order to allow distributions with heavier or lighter tails than the normal ones. Penalized likelihood equations are applied to derive the maximum likelihood estimates which appear to be robust against outlying observations in the sense of the Mahalanobis distance. In order to study the sensitivity of the penalized estimates under some usual perturbation schemes in the model or data, the local influence curvatures are derived and some diagnostic graphics are proposed. Motivating examples preliminary analyzed under normal errors are reanalyzed under some appropriate elliptical errors. The local influence approach is used to compare the sensitivity of the model estimates.

Classical parametric methods of statistical inference and hypothesis testing are derived under fundamental theoretical assumptions, which may or may not be met in real world applications. However, these methods are usually used despite the violation of their underlying assumptions, while it is argued, that these methods are quite insensitive to the violation of relevant assumptions. Moreover, alternative nonparametric or rank tests are often overlooked, mostly because these methods may be deemed to be less powerful then parametric methods. The aim of the dissertation is therefore a description of the consequences of assumption violations concerning classical one-sample and two-sample statistical methods and a consistent and comprehensive comparison of parametric, nonparametric and robust statistical techniques, which is based on extensive simulation study and focused mostly on a normality and heteroscedasticity assumption violation. The results of the simulation study confirmed that the classical parametric methods are relatively robust, with some reservations in case of outlying observations, when traditional methods may fail. On the other hand, the empirical study clearly proved that the classical parametric methods are losing their optimal properties, when the underlying assumptions are violated. For example, in many cases of non-normality the appropriate nonparametric and rank-based methods are more powerful, and therefore a statement, that these methods are unproductive due to their lack of power may be considered a crucial mistake. However, the choice of the most appropriate distribution-free method generally depends on the particular form of the underlying distribution.