## 4 Discussion
Past studies provide evidence for parameter estimation of multivariate distributions only up to four dimensions, and the global optimization technique considered in this study is able to estimate parameters of multivariate distributions with any dimension. The number of parameters to be estimated are ninety-one, twenty-one and one-hundred-and-eighty-three with respect to multivariate distributions associated with AORD, GSFC and ASPI and their related financial markets. It is evident that the parameter estimates provided by the local optimization technique are ambiguous. This characteristic of the local optimization method is prominent due to its main disadvantage of excessive dependence on the initial parameters provided. As a solution, a globaloptimization technique, DSO algorithm, was considered, which does not require initial parameters and is a stable algorithm that provides solid solutions. Results obtained from the global optimization method are much better than the parameters of the multivariate scaled t distributions of return series of AORD, GSPC and ASPI and those of their related financial indices. The discussion below is based on the results obtained from global optimization. The estimates of location parameters are small and around zero, and this implies that the multivariate distribution associated with AORD index and its related financial indices is central. The multivariate scaled t distribution of AORD series and related financial indices is less peaked and have fat tails as the estimated shape parameter is three. The multivariate distribution of GSPC index and related financial indices is symmetric, less peaked and has fat tails as the parameter estimates exhibits similar formation. Furthermore, the multivariate scaled t distribution of ASPI and related indices is central as the location parameters are around zero, less peaked as the shape parameter is three and has fat tails as the shape parameter is closer to unity. The parameter estimates of scale parameter \(\Sigma\) are alike values in each table, and it suggests that the markets have similar influence on the respective multivariate distributions. Incorporating an appropriate multivariate distribution other than the Gaussian distribution as the distribution of input variables to the probabilistic neural network is one of the objectives of the study. Therefore, proposed PNN was implemented using matlab software, which facilitates considering the multivariate scaled t distribution as the distribution of input variables. To evaluate the performance of the proposed PNN, comparisons were carried out with the existing standard PNN implemented using matlab software. For budgetary convenience, the implemented matlab codes can be easily converted into an open source programming language like python. Table 5 illustrates that the overall accuracy levels of standard probabilistic neural network model when forecasting the direction of stock indices AORD, GSPC and ASPI are 54.5%, 53.34% and 55.09%, respectively. Same data sets were used to evaluate the performance of proposed PNN model which consider multivariate scaled t distribution when calculating probabilities in the pattern layer. Overall accuracy of AORD index is 61.56% while GSPC and ASPI indices exhibit 63.04% and 63.47% corresponding accuracy when forecasting the direction using proposed PNN, and this implies an increase in overall accuracy of proposed PNN for all considered stock indices than that of standard PNN. These are acceptable and high percentages of accuracy with respect to financial market forecasting. By referring Table 5, it is noteworthy that the misclassification percentages when applying standard PNN for AORD series for Class 1, Class 2 and Class 3 are 70.59%, 4.67% and 61.25%. Very high misclassification percentages can be observed in Class 1 and Class 3, while a very low misclassification percentage can be observed in Class 2. Consequently, it is evident that the standard PNN model is biased towards one class against the other two classes in this scenario.