Generalized Hyperbolic Distribution for N-Type

Normal distribution is probably the most commonly used distribution in practice. However, due to it symmetric shape, its applicadon in some areas, such as the financial industry, is limited. In a lot of cases, other asymmetric distributions are used such as lognormal distribution. As a much more flexible distribution, generalized hyperbolic distribution can be used for N-type occurrence function. 

Generalized hyperbolic distribution is introduced by Barndorff-Nielsen (1978). Comparing to traditional normal distribution with two parameters p and o, hyperbolic distribution has five parameters X, a, p, 5, and p, and it is much more flexible. The density function is  

In the density function, a determines the shape, P determines the skewness, p determines the location, and A, determines the heaviness of the tails. 5 is the scaling parameter, which is comparable to a in the normal distribution. 

Roughly, about 250 data points are required to fit the generalized hyperbolic distributions. However, about 100 data points can offer reasonable results. Although maximum-likelihood estimation method can be used to estimate the parameters, it is very difficult to solve such a complicated nonlinear equation system with five equations and five unknown parameters. Therefore, numerical algorithms are suggested such as modified Powell method (Wang, 2005). Kolmogorov-Smirnov statistics can also be used here for the fitness test. 

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Taguchi s loss functions represent the impact of the risk. Since the risk is a function of both impact and occurrence, we need the occurrence function of the risk event as well. MtT type occurrence function is actually the distribution of the performance measure from historical data and it can be used as the probability function to predict risk in the future. Firms can use past data to fit an appropriate occurrence function, or use some widely adopted distributions such as Ganuna distribution for S-type occurrence function. Beta distribution for L-type occurrence function, and Generalized Hyperbolic distribution for N-type occurrence function. 

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