Gaussian Approximation of Conditional Elliptical Random Vectors

Asymptotics of the norm of elliptical random vectors

Let \((U_{ni},V_{ni}), 1 \le i \le n, n\ge 1\) be a triangular array of independent bivariate elliptical random vectors with the same distribution function as \(\bigl( S_1, \rho_n S_1 + \sqrt{1- \rho_n^2}S_2\bigr), \rho_n \in (0,1)\) where \((S_1, S_2)\) is a bivariate spherical random vector. Under assumptions on the speed of convergence of \(\rho_n\to 1\) we show in this paper that the maxima of this triangular array is in the max-domain of attraction of a new max-id. distribution function \(H_{\alpha,\lambda}\), provided that \(\sqrt{S_1^2+S_2^2}\) has distribution function in the max-domain of attraction of the Weibull distribution function \(\Psi_\alpha\).

On the residual dependence index of elliptical distributions

Let $(X,Y)$ be a bivariate random vector. The estimation of a probability of the form $P(Y\leq y \mid X >t) $ is challenging when $t$ is large, and a fruitful approach consists in studying, if it exists, the limiting conditional distribution of the random vector $(X,Y)$, suitably normalized, given that $X$ is large. There already exists a wide literature on bivariate models for which this limiting distribution exists. In this paper, a statistical analysis of this problem is done. Estimators of the limiting distribution (which is assumed to exist) and the normalizing functions are provided, as well as an estimator of the conditional quantile function when the conditioning event is extreme. Consistency of the estimators is proved and a functional central limit theorem for the estimator of the limiting distribution is obtained. The small sample behavior of the estimator of the conditional quantile function is illustrated through simulations.

A note on skew-elliptical distributions and linear functions of order statistics

We consider the product of a Wishart matrix and a conditionally elliptical random vector whose dispersion matrix depends on the Wishart matrix. We derive a stochastic representation of the product that does not involve the Wishart matrix and is composed of a chi-squared variable, a singular Gaussian vector, and an elliptical random vector. Consequently, this new representation enables us to avoid the evaluation of integrals over the space of positive semi-definite matrices when deriving the characteristic function, moments, and other distributional quantities of the product. Based on the new representation, we obtain the exact expression for the characteristic function and the first four moments. The kurtosis of the product indicates that asymptotic normality does not hold for large degrees of freedom in the Wishart matrix. Nevertheless, the multivariate Gaussian approximation performs well for large degrees of freedom and large eigenvalues of the covariance matrix of the Wishart matrix. The accuracy of the approximation is numerically investigated by applying it to an elliptical regression model with a stochastic design matrix.

Suppose a multi-Gaussian process is observed at some set of sites, and we wish to obtain the conditional block grade distribution given some observations. We show that this conditional distribution is approximately Gaussian under certain conditions. In particular, given a single observation from a continuous multi-Gaussian process, the conditional distribution under a small change of support is approximately Gaussian unless, roughly speaking, the observed process is twice differentiable and the observation site is at the center of mass of the support region. A Gaussian approximation for the conditional prediction error of the total ore in a fixed region is considered also, although an example demonstrates that a naive analysis can give incorrect limiting conditional means.

We consider an n-player finite strategic game. The payoff vector of each player is a random vector whose distribution is not completely known. We assume that the distribution of a random payoff vector of each player belongs to a distributional uncertainty set. We define a distributionally robust chance-constrained game using worst-case chance constraint. We consider two types of distributional uncertainty sets. We show the existence of a mixed strategy Nash equilibrium of a distributionally robust chance-constrained game corresponding to both types of distributional uncertainty sets. For each case, we show a one-to-one correspondence between a Nash equilibrium of a game and a global maximum of a certain mathematical program.

A Markov tree is a random vector indexed by the nodes of a tree whose distribution is determined by the distributions of pairs of neighbouring variables and a list of conditional independence relations. Upon an assumption on the tails of the Markov kernels associated to these pairs, the conditional distribution of the self-normalized random vector when the variable at the root of the tree tends to infinity converges weakly to a random vector of coupled random walks called a tail tree. If, in addition, the conditioning variable has a regularly varying tail, the Markov tree satisfies a form of one-component regular variation. Changing the location of the root, that is, changing the conditioning variable, yields a different tail tree. When the tails of the marginal distributions of the conditioning variables are balanced, these tail trees are connected by a formula that generalizes the time change formula for regularly varying stationary time series. The formula is most easily understood when the various one-component regular variation statements are tied up into a single multi-component statement. The theory of multi-component regular variation is worked out for general random vectors, not necessarily Markov trees, with an eye towards other models, graphical or otherwise.

Consider the sample X1, X2, ..., XN of N independent and identically distributed (iid) random variables with common cumulative distribution function (cdf)F, and let Fu be their conditional excess distribution function F. We define the ordered sample by . Pickands (1975), Balkema and de Haan (1974) posed that for a large class of underlying distribution functions F , and large u ,Fu is well approximated by the Generalized Pareto Distribution.The mixed method is a method for determining thresholds. This method consists in minimizing the variance of a convex combination of other thresholds.The objective of the mixed method is to determine by which probability distribution one can approach this conditional distribution. In this article, we propose a theorem which specifies the conditional distribution of excesses when the deterministic threshold tends to the end point.

A conditional distribution function-based measure for independence and [formula omitted]-sample tests in multivariate data

Motivated by a (seemingly previously unnoticed) result stating that d −dimensional distributions on $(0,\infty )^{d}$ are characterized by the collection of their min-linear projections, we introduce and study a notion of min-characteristic function (min-CF) of a random vector with strictly positive components. Unlike the related notion of max-characteristic function which has been studied recently, the existence of the min-CF does not hinge on any integrability conditions. It is itself a multivariate distribution function, which is continuous and concave, no matter which properties the initial distribution function has. We show the equivalence between convergence in distribution and pointwise convergence of min-CFs, and we also study the functional convergence of the min-CF of the empirical distribution function of a sample of independent and identically distributed random vectors. We provide some further insight into the structure of the set of min-CFs, and we conclude by showing how transforming the components of an arbitrary random vector by a suitable one-to-one transformation such as the exponential function allows the construction of a notion of min-CF for arbitrary random vectors.

The Multivariate Conditional Stein's Lemma for Elliptical Random Vectors

The need for calculating and characterizing singular normal distributions arises in a natural way when considering chance constraints of the type P(Az <= b(x) >= p, where A is a rectangular matrix having more rows than columns, b is some function and z is a random vector having some nondegenerate multivariate normal distribution. Such situation is typical, for instance, in stochastic networks, where a comparatively small random vector may induce a possibly large number of linear inequality constraints. Passing to the transformed random variable q:=Az, the constraint can be equivalently rewritten as F(b(x))>= p, where F is the distribution function of q. In contrast to the original random vector z, the transformed vector q has a singular normal distribution. The talk demonstrates how to get back from here to (a sum of) regular normal distributions under a full rank regularity condition. This allows for an efficient calculation of singular normal distributions and provides a numerical method which outperforms competing procedures in moderate dimensions. Computational results for test examples are provided for the sake of comparison. In general, if the mentioned regularity condition is violated, then the singular normal distribution function F may even lack continuity. The talk provides an equivalent criterion for Lipschitz continuity of F and characterizes differentiability and subdifferentiability of F.

Strong and Weak Multivariate First-Order Stochastic Dominance