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A Biconvex Form for Copulas

Dependence Modeling

We study the integration of a copula with respect to the probability measure generated by another copula. To this end, we consider the map [. , .] : C × C → R given by [...] where C denotes the collection of all d–dimensional copulas and QD denotes the probability measures associated with the copula D. Specifically, this is of interest since several measures of concordance such as Kendall’s tau, Spearman’s rho and Gini’s gamma can be expressed in terms of the map [. , .]. Quite generally, the map...

A characterization of matrix variate normal distribution.

International Journal of Mathematics and Mathematical Sciences

Metrika

A copula test space model how to avoid the wrong copula choice

Kybernetika

We introduce and discuss the test space problem as a part of the whole copula fitting process. In particular, we explain how an efficient copula test space can be constructed by taking into account information about the existing dependence, and we present a complete overview of bivariate test spaces for all possible situations. The practical use will be illustrated by means of a numerical application based on an illustrative portfolio containing the S&P 500 Composite Index, the JP Morgan Government...

Metrika

A generalization of Wishart density for the case when the inverse of the covariance matrix is a band matrix

Mathematica Bohemica

In a multivariate normal distribution, let the inverse of the covariance matrix be a band matrix. The distribution of the sufficient statistic for the covariance matrix is derived for this case. It is a generalization of the Wishart distribution. The distribution may be used for unbiased density estimation and construction of classification rules.

A geometric approach to correlation inequalities in the plane

Annales de l'I.H.P. Probabilités et statistiques

By elementary geometric arguments, correlation inequalities for radially symmetric probability measures are proved in the plane. Precisely, it is shown that the correlation ratio for pairs of width-decreasing sets is minimized within the class of infinite strips. Since open convex sets which are symmetric with respect to the origin turn out to be width-decreasing sets, Pitt’s Gaussian correlation inequality (the two-dimensional case of the long-standing Gaussian correlation conjecture) is derived...

Metrika

A new family of trivariate proper quasi-copulas

Kybernetika

In this paper, we provide a new family of trivariate proper quasi-copulas. As an application, we show that ${W}^{3}$ – the best-possible lower bound for the set of trivariate quasi-copulas (and copulas) – is the limit member of this family, showing how the mass of ${W}^{3}$ is distributed on the plane $x+y+z=2$ of ${\left[0,1\right]}^{3}$ in an easy manner, and providing the generalization of this result to $n$ dimensions.

A note on Anderson's note on a stationary autoregressive process

Discussiones Mathematicae Probability and Statistics

A form of the covariance matrix of a weakly stationary first-order autoregressive process is established.

A note on biconic copulas

Kybernetika

We describe a class of bivariate copulas having a fixed diagonal section. The obtained class contains both the Fréchet upper and lower bounds and it allows to describe non-trivial tail dependence coefficients along both the diagonals of the unit square.

Metrika

A note on stability of multivariate distributions.

Stochastica

In this note we give an elementary proof of a characterization for stability of multivariate distributions by considering a functional equation.

Metrika

A note on the Galambos copula and its associated Bernstein function

Dependence Modeling

There is an infinite exchangeable sequence of random variables {Xk}k∈ℕ such that each finitedimensional distribution follows a min-stable multivariate exponential law with Galambos survival copula, named after . A recent result of  implies the existence of a unique Bernstein function Ψ associated with {Xk}k∈ℕ via the relation Ψ(d) = exponential rate of the minimum of d members of {Xk}k∈ℕ. The present note provides the Lévy–Khinchin representation for this Bernstein function and explores some...

A short note on multivariate dependence modeling

Kybernetika

As said by Mareš and Mesiar, necessity of aggregation of complex real inputs appears almost in any field dealing with observed (measured) real quantities (see the citation below). For aggregation of probability distributions Sklar designed his copulas as early as in 1959. But surprisingly, since that time only a very few literature have appeared dealing with possibility to aggregate several different pairwise dependencies into one multivariate copula. In the present paper this problem is tackled...

A study of the tangent space model of the von Mises-Fisher distrubution.

RACSAM

For a random rotation X = M0 eφ(ε) where M0 is a 3 x 3 rotation, ε is a trivariate random vector, and φ(ε) is a skew symmetric matrix, the least squares criterion consists of seeking a rotation M called the mean rotation minimizing tr[(M - E(X))t (M - E(X))]. Some conditions on the distribution of ε are set so that the least squares estimator is unbiased. Of interest is when ε is normally distributed N(0;Σ). Unbiasedness of the least squares estimator is dealt with according to eigenvalues of Σ.

A survey on continuous elliptical vector distributions.

Revista Matemática Complutense

In this paper it is taken up a revision and characterization of the class of absolutely continuous elliptical distributions upon a parameterization based on the density function. Their properties (probabilistic characteristics, affine transformations, marginal and conditional distributions and regression) are shown in a practical and easy to interpret way. Two examples are fully undertaken: the multivariate double exponential distribution and the multivariate uniform distribution.

A theoretical argument why the $t$-copula explains credit risk contagion better than the Gaussian copula.

In this paper, we consider a symmetric α-stable p-sub-stable two-dimensional random vector. Our purpose is to show when the function $exp-\left(|a|p+{|b|p\right)}^{\alpha /p}$ is a characteristic function of such a vector for some p and α. The solution of this problem we can find in , in the language of isometric embeddings of Banach spaces. Our proof is based on simple properties of stable distributions and some characterization given in .