Transition Law-based Simulation of Generalized Inverse Gaussian Ornstein–Uhlenbeck Processes

In this paper, a stochastic integral of Ornstein–Uhlenbeck type is represented to be the sum of three independent random variables—one follows a distribution whose density is a deconvolution of the densities of two generalized inverse Gaussian distributions, and the two others all have compound Poisson distributions. Based on the representation of the stochastic integral, a simulation procedure for obtaining discretely observed values of Ornstein–Uhlenbeck processes with given generalized inverse Gaussian distribution is provided. For some subclasses of the generalized inverse Gaussian Ornstein–Uhlenbeck process, the innovations can be sampled exactly. The performance of the simulation method is evidenced by some empirical results.     

An extension of Goldie's result and further results in infinite divisibility

Summary It is shown that gamma random variables satisfy a certain factorisation property. Using this property, Goldie's result on mixtures of the exponential distributions is extended. The note also establishes that the generalized inverse Gaussian distribution and the generalized hyperbolic distribution considered by Barndorff-Nielsen and Halgreen (1977) are indeed self-decomposable.     

Moments of the generalized hyperbolic distribution

In this paper we demonstrate a recursive method for obtaining the moments of the generalized hyperbolic distribution. The method is readily programmable for numerical evaluation of moments. For low order moments we also give an alternative derivation of the moments of the generalized hyperbolic distribution. The expressions given for these moments may be used to obtain moments for special cases such as the hyperbolic and normal inverse Gaussian distributions. Moments for limiting cases such as the skew hyperbolic t and variance gamma distributions can be found using the same approach.     

Two New Mixture Models Related to the Inverse Gaussian Distribution

This article presents a new family of logarithmic distributions to be called the sinh mixture inverse Gaussian model and its associated life distribution referred as the extended mixture inverse Gaussian model. Specifically, the density, distribution function, and moments are developed for the sinh mixture inverse Gaussian distribution. Next, the extended mixture inverse Gaussian distribution is characterized. A graphical analysis of the densities of the new models is also provided. In addition, a lifetime analysis is presented for the extended mixture inverse Gaussian distribution. Finally, an example with a real data set is given to illustrate the methodology, which indicates that the new models result in a better fit to the data than some other well-known distributions.     

The Matsumoto-Yor property and the structure of the Wishart distribution

This paper establishes a link between a generalized matrix Matsumoto-Yor (MY) property and the Wishart distribution. This link highlights certain conditional independence properties within blocks of the Wishart and leads to a new characterization of the Wishart distribution similar to the one recently obtained by Geiger and Heckerman but involving independences for only three pairs of block partitionings of the random matrix.In the process, we obtain two other main results. The first one is an extension of the MY independence property to random matrices of different dimensions. The second result is its converse. It extends previous characterizations of the matrix generalized inverse Gaussian and Wishart seen as a couple of distributions.We present two proofs for the generalized MY property. The first proof relies on a new version of Herz's identity for Bessel functions of matrix arguments. The second proof uses a representation of the MY property through the structure of the Wishart.     

On approximating the renewal function with its linear asymptot: How large is large enough?

A measure of convergence of the renewal function to its linear asymptot is defined and computed for gamma. Weibull, truncated normal, inverse Gaussian and lognormal distributions, using McConalogue's [5] generalized cubic splining algorithm. This measure is then approximated by second degree polynomials in the coefficient of variation.     

Fractal Activity Time Models for Risky Asset with Dependence and Generalized Hyperbolic Distributions

Risky asset models with the dependence through fractal activity time are described. The construction of the fractal activity time is implemented via superpositions of Ornstein-Uhlenbeck type processes driven by Lévy noise. The model features both tractable dependence structure and desired marginal distributions of the returns from the generalized hyperbolic class: the Variance Gamma and normal inverse Gaussian. These distributions provide good fit to real financial data. Pricing formulae for the proposed models are derived.     

A mixture of generalized hyperbolic factor analyzers

The mixture of factor analyzers model, which has been used successfully for the model-based clustering of high-dimensional data, is extended to generalized hyperbolic mixtures. The development of a mixture of generalized hyperbolic factor analyzers is outlined, drawing upon the relationship with the generalized inverse Gaussian distribution. An alternating expectation-conditional maximization algorithm is used for parameter estimation, and the Bayesian information criterion is used to select the number of factors as well as the number of components. The performance of our generalized hyperbolic factor analyzers model is illustrated on real and simulated data, where it performs favourably compared to its Gaussian analogue and other approaches.     

Estimation of coefficient of variation in a weighted inverse Gaussian model

The coefficient of variation is an important parameter in many physical, biological and medical sciences. In this paper we study the estimation of the square of the coefficient of variation in a weighted inverse Gaussian model which is a mixture of the inverse Gaussian and the length biased inverse Gaussian distribution. This represents a rich family of distributions for different values of the mixing parameter and can be used for modelling various life testing situations. The maximum likelihood as well as the Bayes estimates of the parameters are obtained. These estimates are used to derive the estimates of the square of the coefficient of variation of the model under study. Several important data sets are analysed to illustrate the results. © 1996 John Wiley & Sons, Ltd.     

The generalized Cantor distribution and its corresponding inverse distribution

Building on ideas and concepts introduced by Lad, Taylor and Hosking, a generalized Cantor distribution and a corresponding skew generalized Cantor distribution are developed and analyzed. Associated inverse distributions are also introduced. In some cases method of moment estimation is shown to be readily implemented.     

A version of Pitman's 2M — X theorem for geometric Brownian motions

We show that geometric Brownian motion with parameter μ, i.e., the exponential of linear Brownian motion with drift μ, divided by its quadratic variation process is a diffusion process. Taking logarithms and an appropriate scaling limit, we recover the Rogers-Pitman extension to Brownian motion with drift of Pitman's representation theorem for the three-dimensional Bessel process. Time inversion and generalized inverse Gaussian distributions play crucial roles in our proofs.     

Difference equation approaches in evaluation of compound distributions

Panjer (1981), Sundt and Jewell (1981), and Panjer and Willmot (1982) derive computational formulae for the density of the compound distribution when the frequency distribution satisfies certain difference equations. In many instances restrictions are placed on the severity distribution. In this paper it is shown how the results may be adapted to a wider class of severity distributions, including the inverse Gaussian and the Pareto distribution (among others). The computational techniques are clarified and in some cases simplified, and an additional class of frequency distributions is considered, which contains some other well-known distributions. It is then shown how the results may be extended to some well-known contagious frequency distributions such as the Neyman class.     

Subspace clustering for the finite mixture of generalized hyperbolic distributions

The finite mixture of generalized hyperbolic distributions is a flexible model for clustering, but its large number of parameters for estimation, especially in high dimensions, can make it computationally expensive to work with. In light of this issue, we provide an extension of the subspace clustering technique developed for finite Gaussian mixtures to that of generalized hyperbolic distribution. The methodology will be demonstrated with numerical experiments.

     

Properties of the singular, inverse and generalized inverse partitioned Wishart distributions

In this paper we discuss the distributions and independency properties of several generalizations of the Wishart distribution. First, an analog to Muirhead [R.J. Muirhead, Aspects of Multivariate Statistical Theory, Wiley, New York, 1982] Theorem 3.2.10 for the partitioned matrix is established in the case of arbitrary partitioning for singular and inverse Wishart distributions. Second, the density of is derived in the case of singular, non-central singular, inverse and generalized inverse Wishart distributions. The importance of the derived results is illustrated with an example from portfolio theory.     

Simultaneous confidence intervals for several inverse Gaussian populations

In this research, we propose simultaneous confidence intervals for all pairwise comparisons of means from inverse Gaussian distribution. Our method is based on fiducial generalized pivotal quantities for vector parameters. We prove that the constructed confidence intervals have asymptotically correct coverage probabilities. Simulation results show that the simulated Type-I errors are close to the nominal level even for small samples. The proposed approach is illustrated by an example.     

Bivariate inverse gaussian distribution

Summary A bivariate inverse Gaussian (IG) density function is constructed. Relations of the bivariate IG distribution to the normal and χ² distributions are established. The corresponding bivariate random walk (RW) density function is obtained. The properties and behaviour of bivariate IG distribution are studied for large parametric values. Moment estimates of the five parameters are given and applications are pointed out. A generalization to the multivariate IG distribution is proposed.     

Family of multivariate generalized t distributions

In this paper, we introduce a new family of multivariate distributions as the scale mixture of the multivariate power exponential distribution introduced by Gómez et al. (Comm. Statist. Theory Methods 27(3) (1998) 589) and the inverse generalized gamma distribution. Since the resulting family includes the multivariate t distribution and the multivariate generalization of the univariate GT distribution introduced by McDonald and Newey (Econometric Theory 18 (11) (1988) 4039) we call this family as the “multivariate generalized t-distributions family”, or MGT for short. We show that this family of distributions belongs to the elliptically contoured distributions family, and investigate the properties. We give the stochastic representation of a random variable distributed as a multivariate generalized t distribution. We give the marginal distribution, the conditional distribution and the distribution of the quadratic forms. We also investigate the other properties, such as, asymmetry, kurtosis and the characteristic function.     

Goodness-of-Fit Tests for the Inverse Gaussian Distribution Based on the Empirical Laplace Transform

This paper considers two flexible classes of omnibus goodness-of-fit tests for the inverse Gaussian distribution. The test statistics are weighted integrals over the squared modulus of some measure of deviation of the empirical distribution of given data from the family of inverse Gaussian laws, expressed by means of the empirical Laplace transform. Both classes of statistics are connected to the first nonzero component of Neyman's smooth test for the inverse Gaussian distribution. The tests, when implemented via the parametric bootstrap, maintain a nominal level of significance very closely. A large-scale simulation study shows that the new tests compare favorably with classical goodness-of-fit tests for the inverse Gaussian distribution, based on the empirical distribution function.