Some Problems Concerning the Generalized Hyperbolic Distribution

Abstract

Generalized Hyperbolic distribution and its family were recoginized by researchers for the valuable non-Gaussian properties that are applicable in almost all areas of finance and risk management. However due to the complexity of the distribution, there are several problems revealed during implement the distribution to statistical application, in particular the R language. The primary aim of my research is to identify those problems, investigate both theoretical and computational solutions. The basic function that base R provides for most of the classical distribution are probability distribution function (p), density function (d), quantile function (q) and random number generation (r). For generalized hyperbolic distribution and its family, we also followed the same rule. The current estimation approaches for probability distribution function (CDF) and quantile functions of GHyp distribution and skew hyperbolic student’s t distribution which is the limiting case are found to be unstable, inaccurate and lack of efficiency. The problems of GHyp distribution can be solved computationally whilst the problems of the skew hyperbolic student’s t distribution requires proposing a new theoretical method, i.e. the split t transformation method. Besides the problems with CDF and quantile function estimation approach, there are concerns with the current generalized inverse Gaussian distribution random number generation approach which is closely related to GHyp distribution random variates generator. There are few remedies of the concerns has been proposed but not yet implemented include rejection method using either gamma distributed hat function or a two/three parts hat function. These approaches are implemented and compared with the current implemented approaches. Besides the generalized inverse Gaussian distribution, we also investigate the random number generation approach of the hyperbolic distribution which is the sub-case of the GHyp distribution. Apart from the basic functionality of the GHyp distribution, the other prospective of my research was to overhaul the current algorithm and functions for linear modeling using hyperbolic distribution.We also developed a set of functions include but not limited to plot function, summary function in order to have a degree of consistency with the existing model fitting procedures in R. These functions are demonstrated with some real data examples. The algorithm is then compared with several robust modeling techniques using those examples.