Some Problems Concerning the Generalized Hyperbolic and Related Distributions
This thesis presents our attempts to deal with some significant problems concerning the generalized hyperbolic and generalized inverse Gaussian distribution. Firstly, the presence of the modified Bessel function of the second kind Kλ (z) in the density function of these distributions has been known to be one of the challenges for research e ort. Here, the approach is to obtain a reliable algorithm which, for a given /λ/∈ [0,90], can identify a sufficiently large value of Kλ(z )to obtain accurate approximate value of Kλ (z) with a definite number of terms. Consequently, computational and mathematical difficulties which are due to Kλ (z) can be dealt with in a much simpler manner replacing the special function by a finite series. For completeness, a reliable routine to approximate the function when z is small was also implemented. The second problem concerns the fitting of the univariate hyperbolic and univariate generalized inverse Gaussian distribution to data using a numeric optimization algorithm. It is well-known that the log-likelihood functions of these distributions are at and that they need \good" starting values to converge to an optimal value. These problems were reported in the literature even when sample sizes of 500 observations were used. Here, the approaches include obtaining parameter estimates by a symbolic method and reducing the parameter space. The latter is to relax the assumption which must be met for the former to be applicable. Important benefits of the symbolic method are it eliminates the need for starting values and it works stably for sample size of less than 100, or even 20, observations. Thirdly, routines to t subclasses of the multivariate generalized hyperbolic distribution using the Expectation Maximization (EM) algorithm have been derived in the literature. This research suggests an EM based fitting routine for three subclasses of the multivariate generalized hyperbolic, which significantly improves the algorithm convergence speed while maintaining its simplicity. The approach can also be extended to other subclasses. The modified Bessel function of the second kind is one of the important special functions. Research in the literature has identified the evaluation of the incomplete Bessel function as challenging. Thus, significant e ort has been spent on obtaining a numerical method to evaluate it. Here, the problem is approached from both analytical and computational directions and involving three different integral representations of the incomplete Bessel function. Lastly, calculating tail probabilities of the generalized inverse Gaussian has attracted major research e ort. This research utilizes the result obtained from the evaluation of the incomplete Bessel function to derive analytical and numerical approaches to calculate these probabilities.
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