Multivariate Gamma Distributions—<scp>II</scp>

Royen, Thomas

doi:10.1002/0471667196.ess0327.pub2

Cited by 7 publications

(5 citation statements)

References 20 publications

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“…It is worth noting that a recent result of Royen [25] extends this to any two arbitrary symmetric sets, though its full power will not be needed. We refer to the exposition of Latała and Matlak [18].…”

Section: Preliminariesmentioning

confidence: 99%

Vector balancing in Lebesgue spaces

Reis

Rothvoß

2022

Random Struct Algorithms

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The Komlós conjecture suggests that for any vectors bold-italica1,…,bold-italican∈B2m$$ {\boldsymbol{a}}_1,\dots, {\boldsymbol{a}}_n\in {B}_2^m $$ there exist x1,…,xn∈false{prefix−1,1false}$$ {x}_1,\dots, {x}_n\in \left\{-1,1\right\} $$ so that ‖∑i=1nxibold-italicai‖∞≤Ofalse(1false)$$ {\left\Vert {\sum}_{i=1}^n{x}_i{\boldsymbol{a}}_i\right\Vert}_{\infty}\le O(1) $$. It is a natural extension to ask what ℓq$$ {\ell}_q $$‐norm bound to expect for bold-italica1,…,bold-italican∈Bpm$$ {\boldsymbol{a}}_1,\dots, {\boldsymbol{a}}_n\in {B}_p^m $$. We prove a tight partial coloring result for such vectors, implying a nearly tight full coloring bound. As a corollary, this implies a special case of Beck–Fiala's conjecture. We achieve this by showing that, for any δ>0$$ \delta >0 $$, a symmetric convex body K⊆ℝn$$ K\subseteq {\mathbb{R}}^n $$ with Gaussian measure at least eprefix−δn$$ {e}^{-\delta n} $$ admits a partial coloring. Previously this was known only for a small enough δ$$ \delta $$. Additionally, we show that a hereditary volume bound suffices to provide such Gaussian measure lower bounds.

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Section: Preliminariesmentioning

confidence: 99%

Vector balancing in Lebesgue spaces

Reis

Rothvoß

2022

Random Struct Algorithms

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“…The associated multivariate exponential density was discussed earlier by Krishnamoorthy and Parthasarathy (1951), and since then, its properties have been studied by several authors (Krishnaiah and Rao, 1961;Royen, 2004). However, likelihood-based methods for exponential random fields can be troublesome since the analytical expressions of the multivariate density can be derived only in some specific cases.…”

Section: A Random Field With Exponential Marginal Distributionsmentioning

confidence: 99%

Modeling Point Referenced Spatial Count Data: A Poisson Process Approach

Diego

Bevilacqua

Caamaño‐Carrillo

et al. 2022

Journal of the American Statistical Association

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Random fields are useful mathematical tools for representing natural phenomena with complex dependence structures in space and/or time. In particular, the Gaussian random field is commonly used due to its attractive properties and mathematical tractability. However, this assumption seems to be restrictive when dealing with counting data. To deal with this situation, we propose a random field with a Poisson marginal distribution considering a sequence of independent copies of a random field with an exponential marginal distribution as "inter-arrival times" in the counting renewal processes framework. Our proposal can be viewed as a spatial generalization of the Poisson counting process.Unlike the classical hierarchical Poisson Log-Gaussian model, our proposal generates a (non)-stationary random field that is mean square continuous and with Poisson marginal distributions. For the proposed Poisson spatial random field, analytic expressions for the covariance function and the bivariate distribution are provided. In an extensive simulation study, we investigate the weighted pairwise likelihood as a method for estimating the Poisson random field parameters.Finally, the effectiveness of our methodology is illustrated by an analysis of reindeer pellet-group survey data, where a zero-inflated version of the proposed model is compared with zero-inflated Poisson Log-Gaussian and Poisson Gaussian copula models. Supplementary materials for this article, including technical proofs and R code for reproducing the work, are available as an online supplement.

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“…The associated multivariate exponential density was discussed earlier by Krishnamoorthy and Parthasarathy (1951), and its properties have been studied since then by several authors (Krishnaiah and Rao, 1961;Royen, 2004). However, likelihood-based methods for exponential random fields can be troublesome since the analytical expressions of the multivariate density can be derived only in some special cases.…”

Section: A Random Field With Exponential Marginal Distributionsmentioning

confidence: 99%

Modelling Point Referenced Spatial Count Data: A Poisson Process Approach

Diego¹,

Bevilacqua²,

Caamaño‐Carrillo³

et al. 2021

Preprint

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Random fields are useful mathematical tools for representing natural phenomena with complex dependence structures in space and/or time. In particular, the Gaussian random field is commonly used due to its attractive properties and mathematical tractability. However, this assumption seems to be restrictive when dealing with counting data. To deal with this situation, we propose a random field with a Poisson marginal distribution by considering a sequence of independent copies of a random field with an exponential marginal distribution as 'inter-arrival times' in the counting renewal processes framework. Our proposal can be viewed as a spatial generalization of the Poisson process.Unlike the classical hierarchical Poisson Log-Gaussian model, our proposal generates a (non)-stationary random field that is mean square continuous and with Poisson marginal distributions. For the proposed Poisson spatial random field, analytic expressions for the covariance function and the bivariate distribution are provided. In an extensive simulation study, we investigate the weighted pairwise likelihood as a method for estimating the Poisson random field parameters.Finally, the effectiveness of our methodology is illustrated by an analysis of reindeer pellet-group survey data, where a zero-inflated version of the proposed model is compared with zero-inflated Poisson Log-Gaussian and Poisson Gaussian copula models. Supplementary materials for this article, include technical proofs and R code for reproducing the work, are available as an online supplement.

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Multivariate Gamma Distributions—II

Cited by 7 publications

References 20 publications

Vector balancing in Lebesgue spaces

Vector balancing in Lebesgue spaces

Modeling Point Referenced Spatial Count Data: A Poisson Process Approach

Modelling Point Referenced Spatial Count Data: A Poisson Process Approach

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