Approximate Bayesian Computation For Estimation Of Elliptical Stable Distribution And Generalized Extreme-value Distribution

For the financial markets tending to have characteristics of the high-dimensional data and fat tail, if we used multivariate normal distribution to fit it, the result is erroneous. According to financial data feature, we select the Ellipse stable distribution (ESD) to establish a distribution model. But the elliptical stable distribution density did not show the closed form expression, we can't use conventional classical methods (such as maximum likelihood estimation) to estimate the parameters, the paper selects PMC simulation method embedded in Approximate Bayesian computation (ABC) algorithm to estimate the parameters, and employs Summary statistics which is sufficient for the parameters, such as non-diagonal parameters in parameter dispersion matrix concerted projection methods.In addition, we explore the co-movement of stock index among US Dow Jones index, the Shanghai Composite Index and Hong Kong's Hang Seng with the elliptical distribution. For small sample of generalized extreme-value distribution (GEV), we determine four kinds of summary statistics fully reflecting the parameters, implement ABC PMC method to estimate parameter and assess the merits of these summary statistics. Finally, we analyze Beijing and Shanghai's PM2.5 air pollution days with GEV.