Increasing directionally convex orderings of random vectors having the same copula, and their use in comparing ordered data

In this paper, we establish some results for the increasing convex comparisons of generalized order statistics. First, we prove that if the minimum of two sets of generalized order statistics are ordered in the increasing convex order, then the remaining generalized order statistics are also ordered in the increasing convex order. This result is extended to the increasing directionally convex comparisons of random vectors of generalized order statistics. For establishing this general result, we first prove a new result in that two random vectors with a common conditionally increasing copula are ordered in the increasing directionally convex order if the marginals are ordered in the increasing convex order. This latter result is, of course, of interest in its own right.     

A note on the convergence of trivariate extreme order statistics and extension

Necessary and sufficient conditions, under which there exists (at least) a sequence of vectors of real numbers for which the distribution function (d.f.) of any vector of extreme order statistics converges to a nondegenerate limit, are derived. The interesting thing is that these conditions solely depend on the univariate marginals. Moreover, the limit splits into the product of the limit univariate marginals if all the bivariate marginals of the trivariate d.f., from which the sample is drawn, is of negative quadrant dependent random variables (r.v.'s). Finally, all these results are stated for the multivariate extremes with arbitrary dimensions.     

Weak association of random variables

The dependence concept of weak association is introduced and is shown to be equivalent to positive quadrant dependence. Furthermore, a characterization of independence in the class of positive quadrant dependent random variables by means of moment conditions is proved. Both results generalize some theorems proved by Lehmann and Jogdeo for the two- and three-dimensional case.     

Convex combinations of quadrant dependent copulas

It is well known that quadrant dependent (QD) random variables are also quadrant dependent in expectation (QDE). Recent literature has offered examples rigorously establishing the fact that there are QDE random variables which are not QD. The examples are based on convex combinations of specially chosen QD copulas: one negatively QD and another positively QD. In this paper we establish general results that determine when convex combinations of arbitrary QD copulas give rise to negatively or positively QD/QDE copulas. In addition to being an interesting mathematical exercise, the established results are helpful when modeling insurance and financial portfolios.     

On extremal solutions to stochastic control problems,II

The set of attainable laws of the joint state-control process of a controlled diffusion is analyzed from a convex analytic viewpoint. Various equivalence relations depending on one-dimensional marginals thereof are defined on this set and the corresponding equivalence classes are studied.     

How to recognize convexity of a set from its marginals

We investigate the regularity of the marginals onto hyperplanes for sets of finite perimeter. We prove, in particular, that if a set of finite perimeter has log-concave marginals onto a.e. hyperplane then the set is convex. Our proof relies on measuring the perimeter of a set through a Hilbertian fractional Sobolev norm, a fact that we believe has its own interest.     

Markov bases and subbases for bounded contingency tables

In this paper we study the computation of Markov bases for contingency tables whose cell entries have an upper bound. It is known that in this case one has to compute universal Gröbner bases, and this is often infeasible also in small- and medium-sized problems. Here we focus on bounded two-way contingency tables under independence model. We show that when these bounds on cells are positive the set of basic moves of all 2 × 2 minors connects all tables with given margins. We also give some results about bounded incomplete table and we conclude with an open problem on the necessary and sufficient condition on the set of structural zeros so that the set of basic moves of all 2 × 2 minors connects all incomplete contingency tables with given positive margins.     

Adaptive minimax test of independence

The paper is concerned with the adaptive minimax problem of testing the independence of the components of a d-dimensional random vector. The functions under alternatives consist of smooth densities supported on [0, 1]d and separated away from the product of their marginals in L2-norm. We are interested in finding the adaptive minimax rate of testing and a test that attains this rate. We focus mainly on the tests for which the error of the first kind an can decrease to zero as the number of observations increases. We show also how this property of the test affects its error of the second kind.     

Optimality conditions and duality theory for minimizing sums of the largest eigenvalues of symmetric matrices

The sum of the largest eigenvalues of a symmetric matrix is a nonsmooth convex function of the matrix elements. Max characterizations for this sum are established, giving a concise characterization of the subdifferential in terms of a dual matrix. This leads to a very useful characterization of the generalized gradient of the following convex composite function: the sum of the largest eigenvalues of a smooth symmetric matrix-valued function of a set of real parameters. The dual matrix provides the information required to either verify first-order optimality conditions at a point or to generate a descent direction for the eigenvalue sum from that point, splitting a multiple eigenvalue if necessary. Connections with the classical literature on sums of eigenvalues and eigenvalue perturbation theory are discussed. Sums of the largest eigenvalues in the absolute value sense are also addressed.This paper is dedicated to Phil Wolfe on the occasion of his 65th birthday.The work of this author was supported by the National Science Foundation under grants CCR-8802408 and CCR-9101640.The work of this author was supported in part during a visit to Argonne National Laboratory by the Applied Mathematical Sciences subprogram of the Office of Energy Research of the U.S. Department of Energy under contract W-31-109-Eng-38, and in part during a visit to the Courant Institute by the U.S. Department of Energy under Contract DEFG0288ER25053.     

The geometry of certain fixed marginal probability distributions

The geometry of the set of p × q probability mass function matrices with fixed marginals is discussed. The positively quadrant dependent and negatively quadrant dependent subsets are also considered. Explicit graphical representations of these sets are given in the 2 × 2 and 2 × 3 cases.     

Validation of positive quadrant dependence

Quadrant dependence is a useful dependence notion of two random variables, widely applied in reliability, insurance and actuarial sciences. The interest in this dependence structure ranges from modeling it, throughout measuring its strength and investigations on how increasing the dependence effects of several reliability and economic indexes, to hypothesis testing on the dependence. In this paper, we focus on testing for positive quadrant dependence. We propose two new tests for verifying positive quadrant dependence. We prove novel results on finite sample behavior of power function of one of the proposed tests as well as evaluate and compare the two new solutions with the best existing ones, via a simulation study. These comparisons demonstrate that the new solutions are slightly weaker in detecting positive quadrant dependence modeled by classical bivariate models and outperform the best existing solutions when some mixtures, regression and heavy-tailed models have to be detected. Finally, the methods introduced in the paper are applied to real life insurance data, to assess the dependence and test them for positive quadrant dependence.     

Preconditioning Indefinite Systems in Interior Point Methods for Optimization

Every Newton step in an interior-point method for optimization requires a solution of a symmetric indefinite system of linear equations. Most of today's codes apply direct solution methods to perform this task. The use of logarithmic barriers in interior point methods causes unavoidable ill-conditioning of linear systems and, hence, iterative methods fail to provide sufficient accuracy unless appropriately preconditioned. Two types of preconditioners which use some form of incomplete Cholesky factorization for indefinite systems are proposed in this paper. Although they involve significantly sparser factorizations than those used in direct approaches they still capture most of the numerical properties of the preconditioned system. The spectral analysis of the preconditioned matrix is performed: for convex optimization problems all the eigenvalues of this matrix are strictly positive. Numerical results are given for a set of public domain large linearly constrained convex quadratic programming problems with sizes reaching tens of thousands of variables. The analysis of these results reveals that the solution times for such problems on a modern PC are measured in minutes when direct methods are used and drop to seconds when iterative methods with appropriate preconditioners are used.     

Reducing model risk via positive and negative dependence assumptions

We give analytical bounds on the Value-at-Risk and on convex risk measures for a portfolio of random variables with fixed marginal distributions under an additional positive dependence structure. We show that assuming positive dependence information in our model leads to reduced dependence uncertainty spreads compared to the case where only marginals information is known. In more detail, we show that in our model the assumption of a positive dependence structure improves the best-possible lower estimate of a risk measure, while leaving unchanged its worst-possible upper risk bounds. In a similar way, we derive for convex risk measures that the assumption of a negative dependence structure leads to improved upper bounds for the risk while it does not help to increase the lower risk bounds in an essential way. As a result we find that additional assumptions on the dependence structure may result in essentially improved risk bounds.     

A class of models for uncorrelated random variables

We consider the class of multivariate distributions that gives the distribution of the sum of uncorrelated random variables by the product of their marginal distributions. This class is defined by a representation of the assumption of sub-independence, formulated previously in terms of the characteristic function and convolution, as a weaker assumption than independence for derivation of the distribution of the sum of random variables. The new representation is in terms of stochastic equivalence and the class of distributions is referred to as the summable uncorrelated marginals (SUM) distributions. The SUM distributions can be used as models for the joint distribution of uncorrelated random variables, irrespective of the strength of dependence between them. We provide a method for the construction of bivariate SUM distributions through linking any pair of identical symmetric probability density functions. We also give a formula for measuring the strength of dependence of the SUM models. A final result shows that under the condition of positive or negative orthant dependence, the SUM property implies independence.     

On Extensions of the Frank-Wolfe Theorems

In this paper we consider optimization problems defined by a quadratic objective function and a finite number of quadratic inequality constraints. Given that the objective function is bounded over the feasible set, we present a comprehensive study of the conditions under which the optimal solution set is nonempty, thus extending the so-called Frank-Wolfe theorem. In particular, we first prove a general continuity result for the solution set defined by a system of convex quadratic inequalities. This result implies immediately that the optimal solution set of the aforementioned problem is nonempty when all the quadratic functions involved are convex. In the absence of the convexity of the objective function, we give examples showing that the optimal solution set may be empty either when there are two or more convex quadratic constraints, or when the Hessian of the objective function has two or more negative eigenvalues. In the case when there exists only one convex quadratic inequality constraint (together with other linear constraints), or when the constraint functions are all convex quadratic and the objective function is quasi-convex (thus allowing one negative eigenvalue in its Hessian matrix), we prove that the optimal solution set is nonempty.     

Numerically Stable Generation of Correlation Matrices and Their Factors

Correlation matrices—symmetric positive semidefinite matrices with unit diagonal—are important in statistics and in numerical linear algebra. For simulation and testing it is desirable to be able to generate random correlation matrices with specified eigenvalues (which must be nonnegative and sum to the dimension of the matrix). A popular algorithm of Bendel and Mickey takes a matrix having the specified eigenvalues and uses a finite sequence of Givens rotations to introduce 1s on the diagonal. We give improved formulae for computing the rotations and prove that the resulting algorithm is numerically stable. We show by example that the formulae originally proposed, which are used in certain existing Fortran implementations, can lead to serious instability. We also show how to modify the algorithm to generate a rectangular matrix with columns of unit 2-norm. Such a matrix represents a correlation matrix in factored form, which can be preferable to representing the matrix itself, for example when the correlation matrix is nearly singular to working precision.     

High powers of random elements of compact Lie groups

Summary. If a random unitary matrix is raised to a sufficiently high power, its eigenvalues are exactly distributed as independent, uniform phases. We prove this result, and apply it to give exact asymptotics of the variance of the number of eigenvalues of falling in a given arc, as the dimension of tends to infinity. The independence result, it turns out, extends to arbitrary representations of arbitrary compact Lie groups. We state and prove this more general theorem, paying special attention to the compact classical groups and to wreath products. This paper is excerpted from the author's doctoral thesis, [9]. Received: 15 October 1995 / In revised form: 7 March 1996     

Spectral analysis of the fourth moment matrix

The fourth moment of a random vector is a matrix whose elements are all moments of order four which can be obtained from the random vector itself. In this paper, we give a lower bound for its dominant eigenvalue and show that its eigenvectors corresponding to positive eigenvalues are vectorized symmetric matrices. Fourth moments of standardized and exchangeable random vectors are examined in more detail.     

Quantum convex support

Convex support, the mean values of a set of random variables, is central in information theory and statistics. Equally central in quantum information theory are mean values of a set of observables in a finite-dimensional C^∗-algebra A, which we call (quantum) convex support. The convex support can be viewed as a projection of the state space of A and it is a projection of a spectrahedron.Spectrahedra are increasingly investigated at least since the 1990s boom in semi-definite programming. We recall the geometry of the positive semi-definite cone and of the state space. We write a convex duality for general self-dual convex cones. This restricts to projections of state spaces and connects them to results on spectrahedra.Our main result is an analysis of the face lattice of convex support by mapping this lattice to a lattice of orthogonal projections, using natural isomorphisms. The result encodes the face lattice of the convex support into a set of projections in A and enables the integration of convex geometry with matrix calculus or algebraic techniques.