Geometric approach to the skewed normal distribution

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.

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

2016

Dependence Modeling

show abstract

“…, ) T ∈ R ν and whose remaining columns are equal the zero vector. Each such representation can be interpreted as a measure-of-cone representation of a certain skewed ln−ν,p-symmetric distribution generalizing the results for two-and multivariate skewed elliptically contoured distributions in [16] and [34], respectively. Let us emphasize at this point that, on the one hand, the sets (x , .…”

Section: Introductionmentioning

confidence: 77%

“…Measure-of-cone representations of elliptically contoured distributions are derived in [16] and [34] for dimension two and for arbitrary nite dimension, respectively. A representation of more general type is proved in [26] for ln,p-symmetric distributions.…”

Section: Proofs and Discussion Proof Of Theoremmentioning

confidence: 99%

“…A part of the answer is that sometimes di erent representations have di erent numerical and mathematical properties. For another part of the answer, we refer to [16] where it is shown that four di erently motivated approaches to the skewed normal distribution can be uni ed from a geometric point of view. Now, given this uni ed approach, it has the potential property of helping to identify otherwise motivated applications of skewed distributions.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Exact distributions of order statistics from l_n,p-symmetric sample distributions

2017

Dependence Modeling

Abstract:We derive the exact distributions of order statistics from a nite number of, in general, dependent random variables following a joint ln,p-symmetric distribution. To this end, we rst review the special cases of order statistics from spherical as well as from p-generalized Gaussian sample distributions from the literature. To study the case of general ln,p-dependence, we use both single-out and cone decompositions of the events in the sample space that correspond to the cumulative distribution function of the kth order statistic if they are measured by the ln,p-symmetric probability measure. We show that in each case distributions of the order statistics from ln,p-symmetric sample distribution can be represented as mixtures of skewed ln−ν,p-symmetric distributions, ν ∈ { , . . . , n − }.

show abstract

Exact extreme value, product, and ratio distributions under non-standard assumptions

2014

AStA Adv Stat Anal