Main result of this paper is to derive the exact analytical expressions of information and COVARIANCE matrices for multivariate Burr III and logistic distributions. These distributions arise as tractable parametric models in price and income distributions, reliability, economics, Human population, some biological organisms to model agricultural population data and survival data. We showed that all the calculations can be obtained from one main moment multi dimensional integral whose expression is obtained through some particular change of variables. Indeed, we consider that this calculus technique for improper integral has its own importance.

Let S be the matrix of residual sum of square in linear modelY=Ab+e, where the matrix of errors is distributed as elliptically contoured with unknown scale matrix S. For Stein loss function, L1 (S ,S) =tr (SˆS-1) -log |(SˆS-1|-p, and squared loss function, L2 ((SˆS-1) =tr ((SˆS-1-I) 2, we offer empirical Bayes estimators of S, which dominate any scalar multiple of S, i.e., aS, by an effective amount. In fact, this study somehow shows that improvement of the empirical Bayes estimators obtained under the normality assumption remains robust under elliptically contoured model.

Capital portfolio management is considered an important issue in the field of economics and its main subject is about the scientific management of combination choice of assets that meet the specific investment objectives. Maximizing returns and minimizing asset risk are the most important goals in the management of the portfolio of capital. This paper proposes two novel risk measures based on the MLP neural networks and prediction intervals (PI). The MLP based risk is constant and assumes that the uncertainty is uniform in the dataset. The second one is a time varying risk measure that doesn’ t assume uniformity condition. After introducing two novel risk measures, a new cost function is presented to consider the expected returns and the involving risk at the same time. Finally, the COVARIANCE matrix adaptation evolution strategy (CMA-ES) algorithm is used to obtain the optimal portfolio. The validity of the proposed selection process (including risk measures, cost function, and the optimization method) is tested using the dataset of the 18 shares of the Tehran Stock Exchange, and the results are compared with the obtained portfolio using the conditional value at risk (CVaR) criterion as a well-known benchmark.

Multivariate reward processes with reward functions of constant rates, defined  on a semi-Markov process, first were studied by Masuda and Sumita, 1991. Reward processes with nonlinear reward functions were introduced in Soltani, 1996. In this work we study a multivariate process s Z(t) = (Z1 (t),..., Z p (t)), t ≥ 0 , where Z1(t),...,Zp (t) are reward processes with nonlinear reward functions ρ1 ,...,ρ p respectively. The Laplace transform of the COVARIANCE matrix, Σ(t), is specified for given ρ1 ,...,ρ p , and if they are real analytic functions, then the COVARIANCE matrix is fully specified. This result in particular provides an explicit formula for the variances of univariate reward processes. We also view Σ(t) as a solution of a renewal equation.