| The classical work horse in finance and insurance for modeling asset returns is the Gaussian model. However, when modeling complex random phenomena, more flexible distributions are needed which are beyond the normal distribution. This is because most of the financial and economic data are skewed and have "fat tails" due to the presence of outliers. Hence symmetric distributions like normal or others may not be good choices while modeling these kinds of data. Flexible distributions like skew normal distribution allow robust modeling of high-dimensional multimodal and asymmetric data. In this dissertation, we consider a very flexible financial model to construct robust comonotonic lower convex order bounds in approximating the distribution of the sums of dependent log skew normal random variables. The dependence structure of these random variables is based on a recently developed multivariate skew normal distribution, called unified skew normal distribution. In order to accommodate the distribution to the model so considered, we first study inherent properties of this class of skew normal distribution. These properties along with the convex order and comonotonicity of random variables are then used to approximate the distribution function of terminal wealth. By calculating lower bounds in the convex order sense, we consider approximations that reduce the multivariate randomness to univariate randomness. The approximations are used to calculate the risk measure related to the distribution of terminal wealth. The accurateness of the approximation is investigated numerically. Results obtained from our methods are competitive with a more time consuming method called, Monte Carlo method. The dissertation also includes the study of quadratic forms and their distributions in the unified skew normal setting. Regarding the inferential issue of the distribution, we propose an estimation procedure based on the weighted moments approach. Results of our simulations provide an indication of the accuracy of the proposed method. |
