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.     

Robust and bias-corrected estimation of the coefficient of tail dependence

We introduce a robust and asymptotically unbiased estimator for the coefficient of tail dependence in multivariate extreme value statistics. The estimator is obtained by fitting a second order model to the data by means of the minimum density power divergence criterion. The asymptotic properties of the estimator are investigated. The efficiency of our methodology is illustrated on a small simulation study and by a real dataset from the actuarial context.     

Calculus of generalized hyperbolic tetrahedra

We calculate the Jacobian matrix of the dihedral angles of a generalized hyperbolic tetrahedron as functions of edge lengths and find the complete set of symmetries of this matrix.     

The weak asymptotic equivalence and the generalized inverse

In this paper, we discuss the relationship between the weak asymptotic equivalence relation and the generalized inverse in the class \({\mathcal{A}} \) of all nondecreasing and unbounded functions, defined and positive on a half-axis [a,+∞) (a > 0). In the main theorem, we prove a proper characterization of the functional class \({{ORV} \cap \mathcal{A}} \), where ORV is the class of all \({\mathcal{O}} \)-regularly varying functions (in the sense of Karamata).     

Mardia's coefficient of kurtosis in elliptical populations

Mardia (1970) defined a measure of multivariate kurtosis and derived its asymptotic distribution for samples from a multivariate normal population. Some new results on elliptical distributions are used to extend Mardia's results to samples from an elliptical distribution. These results provide a method for testing hypotheses on the kurtosis parameter of elliptical distributions. An appendix provides extensions of Kendall and Stuart's (1977) standard errors of bivariate moments to the third and fourth order.This research was supported by grant DA01070 from the U. S. Public Health Service. Production assistance of Julie Speckart is gratefully acknowledged. Requests for reprints should be sent to: P. M. Bentler, Department of Psychology, University of California, Los Angeles, CA 90024-1563.     

The propagation of weak discontinuities in quasi-linear symmetric hyperbolic systems

Zusammenfassung Die vorliegende Abhandlung untersucht die Fortpflanzung kleiner Unstetigkeiten in Systemen von nichtlinearen hyperbolischen Differentialgleichungen. Ein Ausdruck wird abgeleitet, der die Änderung in der Intensität der Unstetigkeit angibt, wenn diese sich entlang eines Strahls des hyperbolischen Gleichungssystems fortbewegt. Schliesslich wird als Beispiel mit Hilfe des angegebenen Verfahrens die Fortpflanzung von Schallwellen behandelt.     

On the residual dependence index of elliptical distributions

The residual dependence index of bivariate Gaussian distributions is determined by the correlation coefficient. This tail index is of certain statistical importance when extremes and related rare events of bivariate samples with asymptotic independent components are being modeled. In this paper we calculate the partial residual dependence indices of a multivariate elliptical random vector assuming that the associated random radius has distribution function in the Gumbel max-domain of attraction. Furthermore, we discuss the estimation of these indices when the associated random radius possesses a Weibull-tail distribution.     

Estimates on the weak solution of semilinear hyperbolic systems

The Cauchy problem for a semilinear hyperbolic system of the type

Analysis of variance and linear contrasts in experimental design with generalized secant hyperbolic distribution

We consider one-way classification model in experimental design when the errors have generalized secant hyperbolic distribution. We obtain efficient and robust estimators for block effects by using the modified maximum likelihood estimation (MML) methodology. A test statistic analogous to the normal-theory F statistic is defined to test block effects. We also define a test statistic for testing linear contrasts. It is shown that test statistics based on MML estimators are efficient and robust. The methodology readily extends to unbalanced designs.     

The generalized Gompertz distribution

This paper deals with a new generalization of the exponential, Gompertz, and generalized exponential distributions. This distribution is called the generalized Gompertz distribution (GGD). The main advantage of this new distribution is that it has increasing or constant or decreasing or bathtub curve failure rate depending upon the shape parameter. This property makes GGD is very useful in survival analysis. Some statistical properties such as moments, mode, and quantiles are derived. The failure rate function is also derived. The maximum likelihood estimators of the parameters are derived using a simulations study. Real data set is used to determine whether the GGD is better than other well-known distributions in modeling lifetime data or not.     

Tests of elliptical symmetry and the asymptotic tail behavior of the statistics

In this paper, some test statistics Of Kolmogorov type and Cramervon Mises type based on projection pursuit technique are proposed for testing the sphericity problem of a high-dimensional distribution. The limiting distributions of the test statistics are derived under the null hypothesis. The asymptotic properties of Bootstrap approximation are investigated and the tail behaviors of the statistics are studied.     

Inequalities for the generalized trigonometric and hyperbolic functions

The generalized trigonometric functions occur as an eigenfunction of the Dirichlet problem for the one-dimensional p$p$-Laplacian. The generalized hyperbolic functions are defined similarly. Some classical inequalities for trigonometric and hyperbolic functions, such as Mitrinovi?–Adamovi?’s inequality, Lazarevi?’s inequality, Huygens-type inequalities, Wilker-type inequalities, and Cusa–Huygens-type inequalities, are generalized to the case of generalized functions.     

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.     

The generalized linear exponential distribution

This paper deals with a new generalization of the linear exponential distribution. This distribution is called the generalized linear exponential distribution (GLED). Some statistical properties such as moments, modes and quantiles are derived. The failure rate function and the mean residual lifetime are also discussed. The maximum likelihood estimators of the parameters are obtained using a simulation study. Real data are used to determine whether the GLED is better than other well-known distributions in modeling lifetime data or not.     

Bootstrap approximation of tail dependence function

For estimating a rare event via the multivariate extreme value theory, the so-called tail dependence function has to be investigated (see [L. de Haan, J. de Ronde, Sea and wind: Multivariate extremes at work, Extremes 1 (1998) 7-45]). A simple, but effective estimator for the tail dependence function is the tail empirical distribution function, see [X. Huang, Statistics of Bivariate Extreme Values, Ph.D. Thesis, Tinbergen Institute Research Series, 1992] or [R. Schmidt, U. Stadtmüller, Nonparametric estimation of tail dependence, Scand. J. Stat. 33 (2006) 307-335]. In this paper, we first derive a bootstrap approximation for a tail dependence function with an approximation rate via the construction approach developed by [K. Chen, S.H. Lo, On a mapping approach to investigating the bootstrap accuracy, Probab. Theory Relat. Fields 107 (1997) 197-217], and then apply it to construct a confidence band for the tail dependence function. A simulation study is conducted to assess the accuracy of the bootstrap approach.