This note describes estimation algorithms for generalized hyperbolic hyperbolic and nor mal inverse Gaussian distributions These distributions provide a better t to empirically observed log return distributions of nancial assets than the classical normal distributions Based on the better t to the semi heavy tails of nancial assets we can compute more realistic Value at Risk estimates The modell...

In this document the generalized hyperbolic (GH) distribution is explained to the extent which is required to understand the internals of the R package ghyp. Essentially, the density and moment generating functions of the GH distribution and its special cases are provided together with some important properties of the GH family. In addition the algorithm underlying the fitting procedure for the...

Generalized variance is applied for determination of dispersion in a multivariate population and is a successful measure for concentration of multivariate data. In this article, we consider constructing confidence interval and testing the hypotheses about generalized variance in a multivariate normal distribution and give a computational approach. Simulation studies are performed to compare thi...

The multivariate Student-t copula family is used in statistical finance and other areas when there is tail dependence in the data. It often is a good-fitting copula but can be improved on when there is tail asymmetry. Multivariate skew-t copula families can be considered when there is tail dependence and tail asymmetry, and we show how a fast numerical implementation for maximum likelihood esti...

Time-changed Brownian motions are extensively applied as mathematical models for asset returns in Finance. Time change is interpreted as a switch to trade-related business time, different from calendar time. Time-changed Brownian motions can be generated by infinite divisible normal mixtures. The standard multivariate normal mean variance mixtures assume a common mixing variable. This correspon...

Ramachandran (1969, Theorem 8) has shown that for any univariate infinitely divisible distribution and any positive real number α, an absolute moment of order α relative to the distribution exists (as a finite number) if and only if this is so for a certain truncated version of the corresponding Lévy measure. A generalized version of this result in the case of multivariate infinitely divisible ...

Ramachandran (1969) [9, Theorem 8] has shown that for any univariate infinitely divisible distribution and any positive real number α, an absolute moment of order α relative to the distribution exists (as a finite number) if and only if this is so for a certain truncated version of the corresponding Lévy measure. A generalized version of this result in the case of multivariate infinitely divisi...