In this thesis, we have studied L-moments and trimmed L-moments (TL-moments) which are both linear functions of order statistics. We have derived expressions for exact variances and covariances of sample L-moments and of sample TL-moments for any sample size n in terms of first and second-order moments of order statistics from small conceptual sample sizes, which do not depend on the actual sample size n. Moreover, we have established a theorem which characterises the normal distribution in terms of these second-order moments and the characterisation suggests a new test of normality. We have also derived a method of estimation based on TL-moments which gives zero weight to extreme observations. TL-moments have certain advantages over L-moments and method of moments. They exist whether or not the mean exists (for example the Cauchy distribution) and they are more robust to the presence of outliers. Also, we have investigated four methods for estimating the parameters of a symmetric lambda distribution: maximum likelihood method in the case of one parameter and L-moments, LQ-moments and TL-moments in the case of three parameters. The L-moments and TL-moments estimators are in closed form and simple to use, while numerical methods are required for the other two methods, maximum likelihood and LQ-moments. Because of the flexibility and the simplicity of the lambda distribution, it is useful in fitting data when, as is often the case, the underlying distribution is unknown. Also, we have studied the symmetric plotting position for quantile plot assuming a symmetric lambda distribution and conclude that the choice of the plotting position parameter depends upon the shape of the distribution. Finally, we propose exponentially weighted moving average (EWMA) control charts to monitor the process mean and dispersion using the sample L-mean and sample L-scale and charts based on trimmed versions of the same statistics. The proposed control charts limits are less influenced by extreme observations than classical EWMA control charts, and lead to tighter limits in the presence of out-of-control observations.

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Geometric trimming of a surface involves removal of unwanted portions of the surface and providing a new mathematical description for the trimmed patch. This entails creating a new geometry for the trimmed patch, which closely approximates the corresponding portion on the original patch. The procedure is shown to involve obtaining data points on the B-spline surface that lie within the region specified by the parameter values for the trimming curve and describing a new surface which interpolates this new set of data points. This research looks at optimizing the procedure described above by basing the choice of parameter values for the trimming curve, at points where curvature optima occur over the surface. A visualization tool kit has been developed using OpenGL, as a means to discern the difference between the two surfaces. In order to quantify and aid in minimizing the error (difference) in approximating the original surface with the trimmed patch, an error measurement tool developed in MATLAB has been employed. / Master of Science

In this thesis, we study the least trimmed squares artificial neural networks (LTS-ANNs), which are generalization of the least trimmed squares (LTS) estimators frequently used in robust linear parametric regression problems to nonparametric artificial neural networks (ANNs) used for nonlinear regression problems. Two training algorithms are proposed in this thesis. The first algorithm is the incremental gradient descent algorithm. In order to speed up the convergence, the second training algorithm is proposed based on recursive least squares (RLS). Three illustrative examples are provided to test the performances of robustness against outliers for the classical ANNs and the LTS-ANNs. Simulation results show that upon proper selection of the trimming constant of the learning machines, LTS-ANNs are quite robust against outliers compared with the classical ANNs.

The silhouette score is a widely used technique to evaluate the quality of a clustering result. One of the current issues with the silhouette score is its sensitivity to outliers, which can lead to misleading interpretations. This problem is caused by the silhouette score using the arithmetic mean to calculate the average intra and inter-cluster distances. To address this issue, three modified silhouette scores are presented: GenSil, TrimSil, and extended TrimSil, which replace the arithmetic mean with the generalized mean, the trimmed mean and a modified trimmed mean, respectively. Experiments on both simulated and real-world datasets show that GenSil is the most effective method, significantly reducing the impact of outliers and achieving high silhouette scores with negative parameter values. TrimSil also improves silhouette scores but performs worse than GenSil, while the extended TrimSil outperforms TrimSil but is still less effective than GenSil. To further aid in selecting the optimal number of clusters with these modified silhouette scores, a more straightforward visualization technique, the silhouette-parameter plot, is also introduced. / Thesis / Master of Science (MSc)

Na década de 1990, emergiu uma numerosa literatura abordando os efeitos da confiança interpessoal no crescimento econômico dos países. Teoricamente, a confiança afeta o crescimento econômico por afetar as decisões que envolvem incerteza acerca das ações futuras de outros agentes, como: investimentos, contratações de trabalhadores, inovação, dentre outras. Este trabalho utiliza a metodologia corrente nesta literatura, avaliando o papel da confiança no crescimento econômico em um cross section de países para três períodos, utilizando informações, principalmente, das Penn World Tables, World Values Survey e dados de educação da UNESCO. Aplicando a técnica de least trimmed squares é avaliada a robustez da variável confiança quando se retiram observações aberrantes. Encontra-se que a confiança tem um efeito considerável no crescimento econômico, mesmo quando outliers são removidos. Também são realizados exercícios para a correção de possíveis problemas de endogeneidade da variável de confiança. Além disso, o trabalho analisa os determinantes da confiança individual, utilizando um modelo probit cujas variáveis explicativas são: renda, escolaridade, idade, país, religião, dentre outras. Este exercício também é feito para analisar o caso brasileiro. Encontra-se que a confiança é uma variável que depende mais da sociedade ou do grupo que das características individuais e, para o caso brasileiro, verificou-se que independentemente de gênero, escolaridade ou renda, as pessoas não confiam nos demais. / In the 1990\'s a large number of works came out investigating the effects of interpersonal trust on the economic growth of countries. Theoretically, trust affects economic growth by affecting all decisions that involve uncertainty on future actions of other agents, such as: investments, hire of employees, innovation, among others. This study uses the current literature methodology, tackling the trust importance for economic growth on a cross section of countries for three periods, using informations mainly from the Penn World Tables, World Values Survey and educational data from UNESCO. Applying the least trimmed squares technique it is evaluated the robustness of the trust variable when influential observations are excluded. It is found a remarkable estimated effect of trust on economic growth, even when outliers are removed. Also some studies are made in order to correct for possible endogeneity problems of the trust variable. Moreover, the work analyses the determinants of individual trust, using a probit model with the regressors: income, schooling, age, country, religion, among others. This analysis is also applied for the brazilian case. It is found that trust depends more on the society or group than on individual characteristics and, for the brazilian case, it was observed that, no matter which gender, schooling or income level the person belongs to, people do not trust each other.

Na década de 1990, emergiu uma numerosa literatura abordando os efeitos da confiança interpessoal no crescimento econômico dos países. Teoricamente, a confiança afeta o crescimento econômico por afetar as decisões que envolvem incerteza acerca das ações futuras de outros agentes, como: investimentos, contratações de trabalhadores, inovação, dentre outras. Este trabalho utiliza a metodologia corrente nesta literatura, avaliando o papel da confiança no crescimento econômico em um cross section de países para três períodos, utilizando informações, principalmente, das Penn World Tables, World Values Survey e dados de educação da UNESCO. Aplicando a técnica de least trimmed squares é avaliada a robustez da variável confiança quando se retiram observações aberrantes. Encontra-se que a confiança tem um efeito considerável no crescimento econômico, mesmo quando outliers são removidos. Também são realizados exercícios para a correção de possíveis problemas de endogeneidade da variável de confiança. Além disso, o trabalho analisa os determinantes da confiança individual, utilizando um modelo probit cujas variáveis explicativas são: renda, escolaridade, idade, país, religião, dentre outras. Este exercício também é feito para analisar o caso brasileiro. Encontra-se que a confiança é uma variável que depende mais da sociedade ou do grupo que das características individuais e, para o caso brasileiro, verificou-se que independentemente de gênero, escolaridade ou renda, as pessoas não confiam nos demais. / In the 1990\'s a large number of works came out investigating the effects of interpersonal trust on the economic growth of countries. Theoretically, trust affects economic growth by affecting all decisions that involve uncertainty on future actions of other agents, such as: investments, hire of employees, innovation, among others. This study uses the current literature methodology, tackling the trust importance for economic growth on a cross section of countries for three periods, using informations mainly from the Penn World Tables, World Values Survey and educational data from UNESCO. Applying the least trimmed squares technique it is evaluated the robustness of the trust variable when influential observations are excluded. It is found a remarkable estimated effect of trust on economic growth, even when outliers are removed. Also some studies are made in order to correct for possible endogeneity problems of the trust variable. Moreover, the work analyses the determinants of individual trust, using a probit model with the regressors: income, schooling, age, country, religion, among others. This analysis is also applied for the brazilian case. It is found that trust depends more on the society or group than on individual characteristics and, for the brazilian case, it was observed that, no matter which gender, schooling or income level the person belongs to, people do not trust each other.

Master of Science / Department of Statistics / Paul Nelson / Trimmed means are robust estimators of location for distributions having heavy tails. Theory and simulation indicate that little efficiency is lost under normality when using appropriately trimmed means and that their use with data from distributions with heavy tails can result in improved performance. This report uses the principle of equivariance applied to trimmed means sampled from a Cauchy distribution to form a discrepancy function of the data and parameters whose distribution is free of the unknown median and scale parameter. Quantiles of this discrepancy function are estimated via asymptotic normality and simulation and used to construct confidence intervals for the median of a Cauchy distribution. A nonparametric approach based on the distribution of order statistics is also used to construct confidence intervals. The performance of these intervals in terms of coverage rate and average length is investigated via simulation when the data are actually sampled from a Cauchy distribution and when sampling is from normal and logistic distributions. The intervals based on simulation estimation of the quantiles of the discrepancy function are shown to perform well across a range of sample sizes and trimming proportions when the data are actually sampled from a Cauchy distribution and to be relatively robust when sampling is from the normal and logistic distributions.

Master of Science / Department of Statistics / Weixin Yao / In this report, we investigate a robust estimation of the number of components in the mixture of regression models using trimmed information criterion. Compared to the traditional information criterion, the trimmed criterion is robust and not sensitive to outliers. The superiority of the trimmed methods in comparison with the traditional information criterion methods is illustrated through a simulation study. A real data application is also used to illustrate the effectiveness of the trimmed model selection methods.

Master of Science / Department of Statistics / Weixin Yao / Mixtures of factor analyzers have been popularly used to cluster the high dimensional data. However, the traditional estimation method is based on the normality assumptions of random terms and thus is sensitive to outliers. In this article, we introduce a robust estimation procedure of mixtures of factor analyzers using the trimmed likelihood estimator (TLE). We use a simulation study and a real data application to demonstrate the robustness of the trimmed estimation procedure and compare it with the traditional normality based maximum likelihood estimate.