Reverse triangle inequality. Antinorms and semi-antinorms

2016

Dependence Modeling

A measure-of-cone representation of skewed continuous ln,p-symmetric distributions, n ∈ N, p > , is proved following the geometric approach known for elliptically contoured distributions. On this basis, distributions of extreme values of n dependent random variables are derived if the latter follow a joint continuous ln,p-symmetric distribution. Light, heavy, and extremely far tails as well as tail indices are discussed, and new parameters of multivariate tail behavior are introduced.

Section: Preliminariesmentioning

confidence: 99%

Extreme value distributions for dependent jointly l_n,p-symmetrically distributed random variables

Müller

2016

Dependence Modeling

“…For the latter notion, we refer to (Moszyńska and Richter 2012). Figure 2 (drawn with MATLAB, as Figure 1) shows (2, 1)-circles S 2,1 (r) according to different values of the (2, 1)-generalized radius r, starting from a local (central) and turning to more global view.…”

Section: Introductionmentioning

confidence: 99%

The Class of (p,q)-spherical Distributions with an Extension of the Sector and Circle Number Functions

2017

Risks

Self Cite

Abstract:For evaluating the probabilities of arbitrary random events with respect to a given multivariate probability distribution, specific techniques are of great interest. An important two-dimensional high risk limit law is the Gauss-exponential distribution whose probabilities can be dealt with based on the Gauss-Laplace law. The latter will be considered here as an element of the newly-introduced family of (p, q)-spherical distributions. Based on a suitably-defined non-Euclidean arc-length measure on (p, q)-circles, we prove geometric and stochastic representations of these distributions and correspondingly distributed random vectors, respectively. These representations allow dealing with the new probability measures similarly to with elliptically-contoured distributions and more general homogeneous star-shaped ones. This is demonstrated by the generalization of the Box-Muller simulation method. In passing, we prove an extension of the sector and circle number functions.

“…A function g : R n → [0, ∞) which is continuous, positively homogeneous, non-degenerate and superadditive in some fan is called an antinorm in [17]. Thereby, g is called superadditive in a sector C or in the fan F if it satisfies the reverse triangle inequality in C or in every sector of the fan F, respectively.…”

Section: Antinorm-contoured Distributionsmentioning

confidence: 99%

Estimation of Star-Shaped Distributions

Liebscher

2016

Risks

Self Cite

Scatter plots of multivariate data sets motivate modeling of star-shaped distributions beyond elliptically contoured ones. We study properties of estimators for the density generator function, the star-generalized radius distribution and the density in a star-shaped distribution model. For the generator function and the star-generalized radius density, we consider a non-parametric kernel-type estimator. This estimator is combined with a parametric estimator for the contours which are assumed to follow a parametric model. Therefore, the semiparametric procedure features the flexibility of nonparametric estimators and the simple estimation and interpretation of parametric estimators. Alternatively, we consider pure parametric estimators for the density. For the semiparametric density estimator, we prove rates of uniform, almost sure convergence which coincide with the corresponding rates of one-dimensional kernel density estimators when excluding the center of the distribution. We show that the standardized density estimator is asymptotically normally distributed. Moreover, the almost sure convergence rate of the estimated distribution function of the star-generalized radius is derived. A particular new two-dimensional distribution class is adapted here to agricultural and financial data sets.