Minimal metrics in a space of random vectors with fixed one-dimensional marginal distributions

1986 ◽  
Vol 34 (2) ◽  
pp. 1542-1555 ◽  
Author(s):  
S. T. Rachev
Keyword(s):  
Random Vectors ◽  
Marginal Distributions ◽  
One Dimensional ◽  
Minimal Metrics

Dependent Random Variables with Independent Subsets - II

1990 ◽  
Vol 33 (1) ◽  
pp. 24-28 ◽  
Author(s):  
Y. H. Wang
Keyword(s):  
Joint Distribution ◽  
Random Variables ◽  
Marginal Distributions ◽  
Dependent Random Variables ◽  
One Dimensional ◽  
The One

AbstractIn this paper, we consolidate into one two separate problems - dependent random variables with independent subsets and construction of a joint distribution with given marginals. Let N = {1,2,3,...} and X = {Xn; n ∊ N} be a sequence of random variables with nondegenerate one-dimensional marginal distributions {Fn; n ∊ N}. An example is constructed to show that there exists a sequence of random variables Y = {Yn; n ∊ N} such that the components of a subset of Y are independent if and only if its size is ≦ k, where k ≧ 2 is a prefixed integer. Furthermore, the one-dimensional marginal distributions of Y are those of X.

scholarly journals Estimating Differential Entropy using Recursive Copula Splitting

Entropy ◽  
2020 ◽  
Vol 22 (2) ◽  
pp. 236 ◽  
Author(s):  
Gil Ariel ◽  
Yoram Louzoun
Keyword(s):  
Compact Support ◽  
Mixed Type ◽  
Random Variables ◽  
Differential Entropy ◽  
High Dimensions ◽  
Marginal Distributions ◽  
Numerical Examples ◽  
One Dimensional ◽  
Different Dimensions

A method for estimating the Shannon differential entropy of multidimensional random variables using independent samples is described. The method is based on decomposing the distribution into a product of marginal distributions and joint dependency, also known as the copula. The entropy of marginals is estimated using one-dimensional methods. The entropy of the copula, which always has a compact support, is estimated recursively by splitting the data along statistically dependent dimensions. The method can be applied both for distributions with compact and non-compact supports, which is imperative when the support is not known or of a mixed type (in different dimensions). At high dimensions (larger than 20), numerical examples demonstrate that our method is not only more accurate, but also significantly more efficient than existing approaches.

scholarly journals DEPENDENCE, DISPERSIVENESS, AND MULTIVARIATE HAZARD RATE ORDERING

2005 ◽  
Vol 19 (4) ◽  
pp. 427-446 ◽  
Author(s):  
Baha-Eldin Khaledi ◽  
Subhash Kochar
Keyword(s):  
Multivariate Analysis ◽  
Hazard Rate ◽  
Stochastic Order ◽  
Random Vectors ◽  
Positive Dependence ◽  
Marginal Distributions ◽  
Dispersive Ordering ◽  
Hazard Rate Ordering

To compare two multivariate random vectors of the same dimension, we define a new stochastic order called upper orthant dispersive ordering and study its properties. We study its relationship with positive dependence and multivariate hazard rate ordering as defined by Hu, Khaledi, and Shaked (Journal of Multivariate Analysis, 2002). It is shown that if two random vectors have a common copula and if their marginal distributions are ordered according to dispersive ordering in the same direction, then the two random vectors are ordered according to this new upper orthant dispersive ordering. Also, it is shown that the marginal distributions of two upper orthant dispersive ordered random vectors are also dispersive ordered. Examples and applications are given.

scholarly journals Marginal Distributions of Random Vectors Generated by Affine Transformations of Independent Two-Piece Normal Variables

2012 ◽  
Vol 2012 ◽  
pp. 1-10
Author(s):  
Maximiano Pinheiro
Keyword(s):  
Probability Density ◽  
Distribution Functions ◽  
Joint Density ◽  
Cumulative Distribution ◽  
Marginal Probability ◽  
Affine Transformations ◽  
Skewed Distributions ◽  
Random Vectors ◽  
Marginal Distributions ◽  
Cumulative Distribution Functions

Marginal probability density and cumulative distribution functions are presented for multidimensional variables defined by nonsingular affine transformations of vectors of independent two-piece normal variables, the most important subclass of Ferreira and Steel's general multivariate skewed distributions. The marginal functions are obtained by first expressing the joint density as a mixture of Arellano-Valle and Azzalini's unified skew-normal densities and then using the property of closure under marginalization of the latter class.

scholarly journals The Pareto Copula, Aggregation of Risks, and the Emperor's Socks

2008 ◽  
Vol 45 (1) ◽  
pp. 67-84 ◽  
Author(s):  
Claudia Klüppelberg ◽  
Sidney I. Resnick
Keyword(s):  
Empirical Processes ◽  
Asymptotic Limit ◽  
Multivariate Distribution ◽  
Marginal Distributions ◽  
One Dimensional ◽  
Limit Distributions

The copula of a multivariate distribution is the distribution transformed so that one-dimensional marginal distributions are uniform. We review a different transformation of a multivariate distribution which yields standard Pareto for the marginal distributions, and we call the resulting distribution the Pareto copula. Use of the Pareto copula has a certain claim to naturalness when considering asymptotic limit distributions for sums, maxima, and empirical processes. We discuss implications for aggregation of risk and offer some examples.

scholarly journals Computable Bounds on the Spectral Gap for Unreliable Jackson Networks

2015 ◽  
Vol 47 (02) ◽  
pp. 402-424 ◽  
Author(s):  
Paweł Lorek ◽  
Ryszard Szekli
Keyword(s):  
Convergence Rates ◽  
Spectral Gap ◽  
Service Rate ◽  
Spectral Gaps ◽  
Birth And Death Processes ◽  
Marginal Distributions ◽  
Hazard Functions ◽  
Jackson Networks ◽  
One Dimensional ◽  
The One

The goal of this paper is to identify exponential convergence rates and to find computable bounds for them for Markov processes representing unreliable Jackson networks. First, we use the bounds of Lawler and Sokal (1988) in order to show that, for unreliable Jackson networks, the spectral gap is strictly positive if and only if the spectral gaps for the corresponding coordinate birth and death processes are positive. Next, utilizing some results on birth and death processes, we find bounds on the spectral gap for network processes in terms of the hazard and equilibrium functions of the one-dimensional marginal distributions of the stationary distribution of the network. These distributions must be in this case strongly light-tailed, in the sense that their discrete hazard functions have to be separated from 0. We relate these hazard functions with the corresponding networks' service rate functions using the equilibrium rates of the stationary one-dimensional marginal distributions. We compare the obtained bounds on the spectral gap with some other known bounds.

scholarly journals The Pareto Copula, Aggregation of Risks, and the Emperor's Socks

2008 ◽  
Vol 45 (01) ◽  
pp. 67-84 ◽  
Author(s):  
Claudia Klüppelberg ◽  
Sidney I. Resnick
Keyword(s):  
Empirical Processes ◽  
Asymptotic Limit ◽  
Multivariate Distribution ◽  
Marginal Distributions ◽  
One Dimensional ◽  
Limit Distributions

The copula of a multivariate distribution is the distribution transformed so that one-dimensional marginal distributions are uniform. We review a different transformation of a multivariate distribution which yields standard Pareto for the marginal distributions, and we call the resulting distribution the Pareto copula. Use of the Pareto copula has a certain claim to naturalness when considering asymptotic limit distributions for sums, maxima, and empirical processes. We discuss implications for aggregation of risk and offer some examples.

scholarly journals Computable Bounds on the Spectral Gap for Unreliable Jackson Networks

2015 ◽  
Vol 47 (2) ◽  
pp. 402-424 ◽  
Author(s):  
Paweł Lorek ◽  
Ryszard Szekli
Keyword(s):  
Convergence Rates ◽  
Spectral Gap ◽  
Service Rate ◽  
Spectral Gaps ◽  
Birth And Death Processes ◽  
Marginal Distributions ◽  
Hazard Functions ◽  
Jackson Networks ◽  
One Dimensional ◽  
The One

The goal of this paper is to identify exponential convergence rates and to find computable bounds for them for Markov processes representing unreliable Jackson networks. First, we use the bounds of Lawler and Sokal (1988) in order to show that, for unreliable Jackson networks, the spectral gap is strictly positive if and only if the spectral gaps for the corresponding coordinate birth and death processes are positive. Next, utilizing some results on birth and death processes, we find bounds on the spectral gap for network processes in terms of the hazard and equilibrium functions of the one-dimensional marginal distributions of the stationary distribution of the network. These distributions must be in this case strongly light-tailed, in the sense that their discrete hazard functions have to be separated from 0. We relate these hazard functions with the corresponding networks' service rate functions using the equilibrium rates of the stationary one-dimensional marginal distributions. We compare the obtained bounds on the spectral gap with some other known bounds.

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