Analytics of Asymmetry and Transformation to Multivariate Normality Through Copula Functions with Applications in Biomedical Sciences and Finance

In this work, we study and develop certain aspects of the analytics of asymmetry for univariate and multivariate data. Accordingly, the above work consists of three separate parts. In the first part of our work, we introduce a new approach to measure the univariate and multivariate skewness based on quantiles and the properties of odd and even functions. We illustrate through numerous examples and simulations that in the multivariate case the Mardia’s measure of skewness fails to provide consistent and meaningful interpretations. However, our new measure appears to provide an index which is more reasonable. In the second part of our work, our emphasis is to moderate or eliminate asymmetry of multivariate data when the interest is in the study of dependence. Copula transformation has been used as an all-purpose transformation to introduce multivariate normality. Using this approach, even though information about marginal distributions is lost, we are still able to study dependence based modeling problems for asymmetric data using the technique developed for multivariate normal data. We illustrate a variety of applications in areas such as multiple regression, principal component, factor analysis, partial least squares and structural equation models. The results are promising in that our approach shows improvement over results obtained when asymmetry is ignored. The last part of this work is based on the applications of our copula transformation to financial data. Specifically, we consider the problem of estimation of “beta risk” associated with a particular financial asset. Taking S&P500 index as a proxy for market, we suggest three versions of “beta estimates” which are useful in situations when the returns of the assets and market proxy do not have the most ideal probability distribution, namely, bivariate normal or when data may contain some very extreme (high or low) returns. Using the copula based methods, developed earlier in this dissertation, and winsorization, we obtain the estimates which in high skewness scenarios perform better than the traditional least square estimate of market beta