Conditional orderings and positive dependence

Every univariate random variable is smaller, with respect to the ordinary stochastic order and with respect to the hazard rate order, than a right censored version of it. In this paper we attempt to generalize these facts to the multivariate setting. It turns out that in general such comparisons do not hold in the multivariate case, but they do under some assumptions of positive dependence. First we obtain results that compare the underlying random vectors with respect to the usual multivariate stochastic order. A larger slew of results, that yield comparisons of the underlying random vectors with respect to various multivariate hazard rate orders, is given next. Some comparisons with respect to the orthant orders are also discussed.     

Multivariate hazard rate orders

Two multivariate hazard rate stochastic orders are introduced and studied. Their meaning, properties, and relationship to other common stochastic orders are examined and investigated. Some examples that illustrate the theory are detailed. Finally, some applications of the new orders in reliability theory and in actuarial science are described.     

Stochastic comparisons in multivariate mixed model of proportional reversed hazard rate with applications

This paper studies the multivariate mixed proportional reversed hazard rate model having dependent mixing variables. Stochastic comparison as well as aging properties in this model are investigated, and stochastic monotone properties of the population vector with respect to the mixing vector are also discussed. Moreover, MTP₂ dependence among the mixing vectors is proved to imply the increasingness of the reversed hazard rate with respect to the baseline one. Finally, some interesting applications are presented as well.     

Some positive dependence stochastic orders

In this paper we study some stochastic orders of positive dependence that arise when the underlying random vectors are ordered with respect to some multivariate hazard rate stochastic orders, and have the same univariate marginal distributions. We show how the orders can be studied by restricting them to copulae, we give a number of examples, and we study some positive dependence concepts that arise from the new positive dependence orders. We also discuss the relationship of the new orders to other positive dependence orders that have appeared in the literature.     

A multivariate version of Hoeffding’s Phi-Square

A multivariate measure of association is proposed, which extends the bivariate copula-based measure Phi-Square introduced by Hoeffding [22]. We discuss its analytical properties and calculate its explicit value for some copulas of simple form; a simulation procedure to approximate its value is provided otherwise. A nonparametric estimator for multivariate Phi-Square is derived and its asymptotic behavior is established based on the weak convergence of the empirical copula process both in the case of independent observations and dependent observations from strictly stationary strong mixing sequences. The asymptotic variance of the estimator can be estimated by means of nonparametric bootstrap methods. For illustration, the theoretical results are applied to financial asset return data.     

The tail probability of the product of dependent random variables from max-domains of attraction

In this article, we investigate the tail probability of the product of finitely many non-negative dependent random variables. They follow distributions from max-domains of attraction of extreme value distributions and their dependence is modeled via a multivariate Farlie–Gumbel–Morgenstern distribution. For each of the Fréchet, Gumbel and Weibull cases, we obtain an explicit asymptotic formula for the tail probability of the product. Our study extends a few known results in the literature.     

Recovery rates in investment-grade pools of credit assets: A large deviations analysis

We consider the effect of recovery rates on a pool of credit assets. We allow the recovery rate to depend on the defaults in a general way. Using the theory of large deviations, we study the structure of losses in a pool consisting of a continuum of types. We derive the corresponding rate function and show that it has a natural interpretation as the favored way to rearrange recoveries and losses among the different types. Numerical examples are also provided.     

Lower and upper bounds of large deviation for sums of subexponential claims in a multi-risk model

In view of the actual condition of the insurance company, a multi-risk model is proposed. The lower and upper bounds for the sums of subexponential claims in this model are given. The proof method is based on the results of the total claim amount under subexponential class.     

General existence results for reflected BSDE and BSDE

In this paper, we are concerned with the problem of existence of solutions for generalized reflected backward stochastic differential equations (GRBSDEs for short) and generalized backward stochastic differential equations (GBSDEs for short) when the generator is continuous with general growth with respect to the variable y and stochastic quadratic growth with respect to the variable z. We deal with the case of a bounded terminal condition ξ and a bounded barrier L as well as the case of unbounded ones. This is done by using the notion of generalized BSDEs with two reflecting barriers studied in Essaky and Hassani (submitted for publication) [14]. The work is suggested by the interest the results might have in finance, control and game theory.     

A vector multivariate hazard rate

A vector definition of multivariate hazard rate, and associated definitions of increasing and decreasing multivariate hazard rate distributions are presented. Consequences of these definitions are worked out in a number of special cases. Relationships between hazard rate and orthant dependence are established.     

Perturbed and non-perturbed Brownian taboo processes

In this paper we study the Brownian taboo process, which is a version of Brownian motion conditioned to stay within a finite interval, and the α-perturbed Brownian taboo process, which is an analogous version of an α-perturbed Brownian motion.We are particularly interested in the asymptotic behaviour of the supremum of the taboo process, and our main results give integral tests for upper and lower functions of the supremum as t→∞. In the Brownian case these include extensions of recent results in Lambert [4], but are proved in a quite different way.     

Tail order and intermediate tail dependence of multivariate copulas

In order to study copula families that have tail patterns and tail asymmetry different from multivariate Gaussian and t copulas, we introduce the concepts of tail order and tail order functions. These provide an integrated way to study both tail dependence and intermediate tail dependence. Some fundamental properties of tail order and tail order functions are obtained. For the multivariate Archimedean copula, we relate the tail heaviness of a positive random variable to the tail behavior of the Archimedean copula constructed from the Laplace transform of the random variable, and extend the results of Charpentier and Segers [7] [A. Charpentier, J. Segers, Tails of multivariate Archimedean copulas, Journal of Multivariate Analysis 100 (7) (2009) 1521–1537] for upper tails of Archimedean copulas. In addition, a new one-parameter Archimedean copula family based on the Laplace transform of the inverse Gamma distribution is proposed; it possesses patterns of upper and lower tails not seen in commonly used copula families. Finally, tail orders are studied for copulas constructed from mixtures of max-infinitely divisible copulas.     

Asymptotics of weighted empirical processes of linear fields with long-range dependence

We discuss the asymptotic behavior of weighted empirical processes of stationary linear random fields in with long-range dependence. It is shown that an appropriately standardized empirical process converges weakly in the uniform-topology to a degenerated process of the form fZ, where Z is a standard normal random variable and f is the marginal probability density of the underlying random field.     

On testing equality of pairwise rank correlations in a multivariate random vector

Spearman’s rank-correlation coefficient (also called Spearman’s rho) represents one of the best-known measures to quantify the degree of dependence between two random variables. As a copula-based dependence measure, it is invariant with respect to the distribution’s univariate marginal distribution functions. In this paper, we consider statistical tests for the hypothesis that all pairwise Spearman’s rank correlation coefficients in a multivariate random vector are equal. The tests are nonparametric and their asymptotic distributions are derived based on the asymptotic behavior of the empirical copula process. Only weak assumptions on the distribution function, such as continuity of the marginal distributions and continuous partial differentiability of the copula, are required for obtaining the results. A nonparametric bootstrap method is suggested for either estimating unknown parameters of the test statistics or for determining the associated critical values. We present a simulation study in order to investigate the power of the proposed tests. The results are compared to a classical parametric test for equal pairwise Pearson’s correlation coefficients in a multivariate random vector. The general setting also allows the derivation of a test for stochastic independence based on Spearman’s rho.     

Deviations of discrete distributions and a question of Móri

In this note, we consider a question of Móri regarding estimating the deviation of the kth terms of two discrete probability distributions in terms of the supremum distance between their generating functions over the interval [0,1]. An optimal bound for distributions on finite support is obtained. Properties of Chebyshev polynomials are employed.     

Approximation of stationary solutions of Gaussian driven stochastic differential equations

We study sequences of empirical measures of Euler schemes associated to some non-Markovian SDEs: SDEs driven by Gaussian processes with stationary increments. We obtain the functional convergence of this sequence to a stationary solution to the SDE. Then, we end the paper by some specific properties of this stationary solution. We show that, in contrast to Markovian SDEs, its initial random value and the driving Gaussian process are always dependent. However, under an integral representation assumption, we also obtain that the past of the solution is independent of the future of the underlying innovation process of the Gaussian driving process.     

Lower classes of the Riemann–Liouville process

Let be a Riemann–Liouville process with index H>0. We characterize the lower classes of its sup-norm statistic by a unique integral test and thus measure the influence of the non-stationarity of increments.     

Reflected backward stochastic differential equation with jumps and RCLL obstacle

In this paper we study one-dimensional reflected backward stochastic differential equation when the noise is driven by a Brownian motion and an independent Poisson point process when the solution is forced to stay above a right continuous left limits obstacle. We prove existence and uniqueness of the solution by using a penalization method combined with a monotonic limit theorem.     

On fine fractal properties of generalized infinite Bernoulli convolutions

The paper is devoted to the investigation of generalized infinite Bernoulli convolutions, i.e., the distributions μξ of the following random variables: where ak are terms of a given positive convergent series; ξk are independent random variables taking values 0 and 1 with probabilities p0k and p1k correspondingly.We give (without any restriction on {an}) necessary and sufficient conditions for the topological support of ξ to be a nowhere dense set. Fractal properties of the topological support of ξ and fine fractal properties of the corresponding probability measure μξ itself are studied in details for the case where ak?rk:=ak+1+ak+2+? (i.e., rk−1?2rk) for all sufficiently large k. The family of minimal dimensional (in the sense of the Hausdorff–Besicovitch dimension) supports of μξ for the above mentioned case is also studied in details. We describe a series of sets (with additional structural properties) which play the role of minimal dimensional supports of generalized Bernoulli convolutions. We also show how a generalization of M. Cooper's dimensional results on symmetric Bernoulli convolutions can easily be derived from our results.     

Variational Inequalities for Arbitrary Multivariate Distributions

Upper bounds for the total variation distance between two arbitrary multivariate distributions are obtained in terms of the correspondingw-functions. The results extend some previous inequalities satisfied by the normal distribution. Some examples are also given.