On multivariate Gaussian tails

Let {X _{n, n} ≥1} be a sequence of standard Gaussian random vectors in ℝ _d ,d ≥ 2. In this paper we derive lower and upper bounds for the tail probabilityP{X _n >t _n } witht _n ∈ ℝ _d some threshold. We improve for instance bounds on Mills ratio obtained by Savage (1962,J. Res. Nat. Bur. Standards Sect. B,66, 93–96). Furthermore, we prove exact asymptotics under fairly general conditions on bothX _n andt _n , as ‖t _n ‖→∞ where the correlation matrix Σ _n ofX _n may also depend onn.     

A note on characterizations of multivariate stable distributions

Several characterizations of multivariate stable distributions together with a characterization of multivariate normal distributions and multivariate stable distributions with Cauchy marginals are given. These are related to some standard characterizations of marcinkiewicz.Research supported, in part, by the Air Force Office of Scientific Research under Contract AFOSR 84-0113. Reproduction in whole or part is permitted for any purpose of the United States Government.     

A note on the representation of parametric models for multivariate extremes

In this note, the representations of extremal Dirichlet and logistic distributions are reviewed and extended. These new representations allow exact simulations of the spectral distribution functions and an extension of the extremal logistic case to dimensions higher than two.      

A note on the Bayesian inference for generalized multivariate gamma distribution

In this paper, we show that the derivation of Lemma 3 of Das and Dey (2010) needs to be corrected by using a logical transformation, instead of the ad-hoc transformation which is partially motivated by its univariate equivalent transformation. The correct derivation is presented by two approaches.     

A continuation approach to mode-finding of multivariate Gaussian mixtures and kernel density estimates

This paper presents a novel method of multi-objective optimization by learning automata (MOLA) to solve complex multi-objective optimization problems. MOLA consists of multiple automata which perform sequential search in the solution domain. Each automaton undertakes dimensional search in the selected dimension of the solution domain, and each dimension is divided into a certain number of cells. Each automaton performs a continuous search action, instead of discrete actions, within cells. The merits of MOLA have been demonstrated, in comparison with a multi-objective evolutionary algorithm based on decomposition (MOEA/D) and non-dominated sorting genetic algorithm II (NSGA-II), on eleven multi-objective benchmark functions and an optimal problem in the midwestern American electric power system which is integrated with wind power, respectively. The simulation results have shown that MOLA can obtain more accurate and evenly distributed Pareto fronts, in comparison with MOEA/D and NSGA-II.     

Convexity results and sharp error estimates in approximate multivariate integration

An interesting property of the midpoint rule and the trapezoidal rule, which is expressed by the so-called Hermite-Hadamard inequalities, is that they provide one-sided approximations to the integral of a convex function. We establish multivariate analogues of the Hermite-Hadamard inequalities and obtain access to multivariate integration formulae via convexity, in analogy to the univariate case. In particular, for simplices of arbitrary dimension, we present two families of integration formulae which both contain a multivariate analogue of the midpoint rule and the trapezoidal rule as boundary cases. The first family also includes a multivariate analogue of a Maclaurin formula and of the two-point Gaussian quadrature formula; the second family includes a multivariate analogue of a formula by P.C. Hammer and of Simpson's rule. In both families, we trace out those formulae which satisfy a Hermite-Hadamard inequality. As an immediate consequence of the latter, we obtain sharp error estimates for twice continuously differentiable functions.

     

Asymptotics of multivariate conditional risk measures for Gaussian risks

This paper investigates accurate approximations of marginal moment excess, marginal conditional tail moment and marginal moment shortfall for multivariate Gaussian system risks. Based on the dimension reduction property via the quadratic programming problem, the super-exponential and polynomial convergence speeds are specified. Two interesting questions involved in risk management are well addressed, namely the minimal additional risk capital injection to avoid infinite risk contagion and a sufficient and necessary condition to alternate the convergence speeds. Numerical study and typical examples are given to illustrate the efficiency of our findings. Due to the flexible moment order, additional applications may involve in risk management, including tail mean–variance portfolio and multivariate conditional risk measures of tail covariance, tail skewness with dependence and extremal risk contagion under consideration.     

A note on perfect Gaussian elimination

We extend a result of D. J. Rose [9] on perfect Gaussian elimination for symmetric matrices. It is proved that the restriction that all pivots are to be chosen along the main diagonal can be removed without loss of generality.     

A note on minimum distance estimates

It is shown that minimum distance estimates enjoy the invariance property of maximum likelihood estimates. Supported under Grant No. 18-2773 of the American University of Beirut.     

A note on generalized averaged Gaussian formulas

We have recently proposed a very simple numerical method for constructing the averaged Gaussian quadrature formulas. These formulas exist in many more cases than the real positive Gauss–Kronrod formulas. In this note we try to answer whether the averaged Gaussian formulas are an adequate alternative to the corresponding Gauss–Kronrod quadrature formulas, to estimate the remainder term of a Gaussian rule.     

A general multivariate chain ladder model

A general multivariate stochastic reserving model is formulated, which not only specifies contemporaneous correlations, but also allows structural connections among triangles. Its structure extends the existing multivariate chain ladder models in a natural way, and this extension proves to be advantageous in improving model adequacy and increasing model flexibility. It is general in the sense that it includes various models in the chain ladder framework as special cases. At the heart of this model is the seemingly unrelated regression technique, which is utilized to estimate parameters that reflect contemporaneous correlations. The use of this technique is essential to construct flexible models, and related statistical theories are applied to study properties of existing estimators. A numerical example is utilized to show the advantage of the proposed model in studying multiple triangles that are related both structurally and contemporaneously.     

On Piterbarg’s max-discretisation theorem for multivariate stationary Gaussian processes

Let {X(t),t≥0}$\{X\left(t\right),t\ge 0\}$ be a stationary Gaussian process with zero-mean and unit variance. A deep result derived in Piterbarg (2004)  [23], which we refer to as Piterbarg’s max-discretisation theorem gives the joint asymptotic behaviour (T→∞$T\to \infty$) of the continuous time maximum M(T)=_maxt∈[0,T]X(t)$M\left(T\right)={max}_{t\in [0,T]}X\left(t\right)$, and the maximum M^δ(T)=_maxt∈R(δ)X(t)$M^{\delta }\left(T\right)={max}_{t\in \mathfrak{R}\left(\delta \right)}X\left(t\right)$, with R(δ)⊂[0,T]$\mathfrak{R}\left(\delta \right)\subset [0,T]$ a uniform grid of points of distance δ=δ(T)$\delta =\delta \left(T\right)$. Under some asymptotic restrictions on the correlation function Piterbarg’s max-discretisation theorem shows that for the limit result it is important to know the speed δ(T)$\delta \left(T\right)$ approaches 0 as T→∞$T\to \infty$. The present contribution derives the aforementioned theorem for multivariate stationary Gaussian processes.     

A note on pivot size in Gaussian elimination

In the present note it is proved that, given a real n × n matrix A_n=(a_ij), such that |a_ij| ? 1, the maximum values in modulus of the pivots p₃,p₄ in Gaussian elimination with complete pivoting are 2$\text{1}\text{4}$ and 4, and 4, respectively. In addition, it is shown that |p₅| <5.005.     

A note on partial pivoting and Gaussian elimination

Summary The results of Wilkinson on the stability of Gaussian elimination with column pivoting are generalized.     

A multivariate cryptosystem inspired by random linear codes

We introduce a new multivariate encryption scheme inspired by random linear codes. The construction is similar to that of UOV, one of the oldest and most trusted multivariate signature schemes, but with a parameterization nothing like that of UOV. The structure of the scheme admits many generic modifications providing an array of security and performance properties. The scheme also supports an embedding modifier which allows any efficiently invertible multivariate system to be incorporated into the scheme. The product of this methodology is the fastest secure multivariate encryption scheme targeting CCA security at the 128-bit level.     

A note on positive distributions in Gaussian analysis

We describe positive generalized functional in Gaussian analysis. We focus on the large distribution space (N)^–1. It is shown that a positive distribution is represented by a measure with specific growth of its moments. Equivalently, this statement can be replaced by the integrability condition.Dedicated to Yu. M. Berezanskii on the occasion of his 70th birthdayPublished in Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 5, pp. 649–658, May, 1995.     

A note on operator self-similar Gaussian vector fields

In this work, a general representation for an operator self-similar Gaussian vector field is obtained, and some properties are studied. It is also shown that such a process is the operator scaling limit of a Gaussian vector field.