In this work we introduce a novel approach of construction of multivariate cumulative distribution functions, based on cyclical-monotone mapping of an original measure $μ\in \mathcal{P}^{ac}_2(\mathbb{R}^d)$ to some target measure $ν\in \mathcal{P}^{ac}_2(\mathbb{R}^d)$ , supported on a convex compact subset of $\mathbb{R}^d$. This map is referred to as $ν$-Brenier distribution function ($ν$-BDF), whose counterpart under the one-dimensional setting $d = 1$ is an ordinary CDF, with $ν$ selected as $\mathcal{U}[0, 1]$, a uniform distribution on $[0, 1]$. Following one-dimensional frame-work, a multivariate analogue of Glivenko-Cantelli theorem is provided. A practical applicability of the theory is then illustrated by the development of a non-parametric pivotal two-sample test, that is rested on $2$-Wasserstein distance.
A (p-1)-variate integral representation is given for the cumulative distribution function of the general p-variate non-central gamma distribution with a non-centrality matrix of any admissible rank. The real part of products of well known analytical functions is integrated over arguments from (-pi,pi). To facilitate the computation, these formulas are given more detailed for p=2 and p=3. These (p-1)-variate integrals are also derived for the diagonal of a non-central complex Wishart Matrix. Furthermore, some alternative formulas are given for the cases with an associated "one-factorial" (pxp)-…
This paper considers a multivariate spatial random field, with each component having univariate marginal distributions of the skew-Gaussian type. We assume that the field is defined spatially on the unit sphere embedded in $\mathbb{R}^3$, allowing for modeling data available over large portions of planet Earth. This model admits explicit expressions for the marginal and cross covariances. However, the $n$-dimensional distributions of the field are difficult to evaluate, because it requires the sum of $2^n$ terms involving the cumulative and probability density functions of a $n$-dimensional Ga…
Some improved Gaussian correlation inequalities for symmetrical n-rectangles extended to some multivariate gamma distributions and some further probability inequalities
Zero-couplings of infinite measures with cyclically monotone support and multivariate regular variation
Aarhus University（奥胡斯大学）；Université Paris Cité, Université Paris Saclay, ENS Paris Saclay, CNRS, SSA, INSERM, Centre Borelli（巴黎西岱大学、巴黎萨克雷大学、巴黎萨克雷高等师范学院、法国国家科学研究中心、社会卫生与老龄化研究所、博尔塞利中心）
There exist several endeavors proposing a new family of extended distributions using the beta-generating technique. This is a well-known mechanism in developing flexible distributions, by embedding the cumulative distribution function (cdf) of a baseline distribution within the beta distribution that acts as a generator. Univariate beta-generated distributions offer many fruitful and tractable properties and have applications in hydrology, biology and environmental sciences amongst other fields. In the univariate cases, this extension works well, however, for multivariate cases, the beta distr…
This paper provides bifactor gamma distribution, trivariate gamma distribution and two copula families on [0, 1] n obtained from the Laplace transforms of the multivariate gamma distribution and the multi-factor gamma distribution given by [P ($θ$)] --$λ$ and [P ($θ$)] --$λ$ n i=1 (1 + pi$θ$i) --($λ$ i --$λ$) respectively, where P is an affine polynomial with respect to the n variables $θ$1,. .. , $θ$n. These copulas are new generalized Farlie-Gumbel-Morgenstern copulas and allow in particular to obtain multivariate gamma distributions for which the cumulative distribution functions and the pr…
Dependency functions of dependent variables are relevant for i) performing uncertainty quantification and sensitivity analysis in presence of dependent variables and/or correlated variables, and ii) simulating random dependent variables. In this paper, we mathematically derive practical dependency functions for classical multivariate distributions such as Dirichlet, elliptical distributions and independent uniform (resp. gamma and Gaussian) variables under constraints that are ready to be used. Since such dependency models are used for sampling random values and we have many dependency models …
Non-asymptotic approximations for Pearson's chi-square statistic and its application to confidence intervals for strictly convex functions of the probability weights of discrete distributions
Extreme Eigenvalue Distributions of Some Complex Correlated Non-Central Wishart and Gamma-Wishart Random Matrices
令$\mathbf{W}$是通过$\mathbf{W}=\mathbf{X}^H\mathbf{X}$定义的相关复非中心Wishart矩阵，其中$\mathbf{X}$是具有非零均值$\boldsymbolΥ$和非平凡协方差$\boldsymbolΣ$的$n\times m \, (n\geq m)$复高斯。我们导出了极值特征值的累积分布函数（c.d.f.s）的精确表达式（即，最大值和最小值）。这些结果非常简单，涉及快速收敛的无穷级数，并适用于$\boldsymbolΥ$具有秩1的实际重要情况。对于一类伽玛-Wishart随机矩阵，$\boldsymbolΥ^H\boldsymbolΥ$服从矩阵变量Γ分布，我们也得到了类似的结果.本文中的特征值分布在无线通信系统中有各种应用，并出现在其他领域，如计量经济学，统计物理学和多元统计。
