Bivariate distributions via a Pareto conditional distribution and a regression function

Uniqueness of specification of a bivariate distribution by a Pareto conditional and a consistent regression function is investigated. New characterizations of the Mardia bivariate Pareto distribution and the bivariate Pareto conditionals distribution are obtained.     

Improved convex upper bound via conditional comonotonicity

Comonotonicity provides a convenient convex upper bound for a sum of random variables with arbitrary dependence structure. Improved convex upper bound was introduced via conditioning by Kaas et al. [Kaas, R., Dhaene, J., Goovaerts, M., 2000. Upper and lower bounds for sums of random variables. Insurance: Math. Econ. 27, 151-168]. In this paper, we unify these results in a more general context using the concept of conditional comonotonicity. We also construct an approximating sequence of convex upper bounds with nice convergence properties.     

Some conditional expectation identities for the multivariate normal

We give formulas for the conditional expectations of a product of multivariate Hermite polynomials with multivariate normal arguments. These results are extended to include conditional expectations of a product of linear combination of multivariate normals. A unified approach is given that covers both Hermite and modified Hermite polynomials, as well as polynomials associated with a matrix whose eigenvalues may be both positive and negative.     

On Moments of Folded and Truncated Multivariate Normal Distributions

Recurrence relations for integrals that involve the density of multivariate normal distributions are developed. These recursions allow fast computation of the moments of folded and truncated multivariate normal distributions. Besides being numerically efficient, the proposed recursions also allow us to obtain explicit expressions of low-order moments of folded and truncated multivariate normal distributions. Supplementary material for this article is available online.     

基于正态分布的广义Liouville分布

对基于正态分布的广义Liouville分布族的性质和特征进行了研究,包括它的边缘分布,条件分布,特征函数和矩等,并对它们的正态性和独立性以及在回归分析中的应用展开了讨论     

Conditional independence,conditional mixing and conditional association

Some properties of conditionally independent random variables are studied. Conditional versions of generalized Borel-Cantelli lemma, generalized Kolmogorov’s inequality and generalized Hájek-Rényi inequality are proved. As applications, a conditional version of the strong law of large numbers for conditionally independent random variables and a conditional version of the Kolmogorov’s strong law of large numbers for conditionally independent random variables with identical conditional distributions are obtained. The notions of conditional strong mixing and conditional association for a sequence of random variables are introduced. Some covariance inequalities and a central limit theorem for such sequences are mentioned.     

Transformation of grouped data to near normality

Box and Cox (1964) proposed a power transformation which has proven utility for transforming ungrouped data to near normality. In this paper, we extend its applicability to grouped data. Illustrative examples are presented and the asymptotic properties of the estimators derived.     

Newsvendor solutions via conditional value-at-risk minimization

In this paper, we consider the minimization of the conditional value-at-risk (CVaR), a most preferable risk measure in financial risk management, in the context of the well-known single-period newsvendor problem, which is originally formulated as the maximization of the expected profit or the minimization of the expected cost. We show that downside risk measures including the CVaR are tractable in the problem due to their convexity, and consequently, under mild assumptions on the probability distribution of products’ demand, we provide analytical solutions or linear programming (LP) formulation of the minimization of the CVaR measures defined with two different loss functions. Numerical examples are also exhibited, clarifying the difference among the models analyzed in this paper, and demonstrating the efficiency of the LP solutions.     

The Construction of Multivariate Distributions from Markov Random Fields

We address the problem of constructing and identifying a valid joint probability density function from a set of specified conditional densities. The approach taken is based on the development of relations between the joint and the conditional densities using Markov random fields (MRFs). We give a necessary and sufficient condition on the support sets of the random variables to allow these relations to be developed. This condition, which we call the Markov random field support condition, supercedes a common assumption known generally as the positivity condition. We show how these relations may be used in reverse order to construct a valid model from specification of conditional densities alone. The constructive process and the role of conditions needed for its application are illustrated with several examples, including MRFs with multiway dependence and a spatial beta process.     

Multivariate Normal Slice Sampling

By introducing auxiliary variables, the traditional Markov chain Monte Carlo method can be improved in certain cases by implementing a “slice sampler.” In the current literature, this sampling technique is used to sample from multivariate distributions with both single and multiple auxiliary variables. When the latter is employed, it generally updates one component at a time.

In this article, we propose two variations of a new multivariate normal slice sampling method that uses multiple auxiliary variables to perform multivariate updating. These methods are flexible enough to allow for truncation to a rectangular region and/or exclusion of any n-dimensional hyper-quadrant. We present results of our methods and existing state-of-the-art slice samplers by comparing efficiency and accuracy. We find that we can generate approximately iid samples at a rate that is more efficient than other methods that update all dimensions at once. Supplemental materials are available online.     

Optimal bandwidth selection for conditional efficiency measures: A data-driven approach

In productivity analysis an important issue is to detect how external (environmental) factors, exogenous to the production process and not under the control of the producer, might influence the production process and the resulting efficiency of the firms. Most of the traditional approaches proposed in the literature have serious drawbacks. An alternative approach is to describe the production process as being conditioned by a given value of the environmental variables (Cazals, C., Florens, J.P., Simar, L., 2002. Nonparametric Frontier estimation: A robust approach. Journal of Econometrics 106, 1–25; Daraio, C., Simar, L., 2005. Introducing environmental variables in nonparametric Frontier models: A probabilistic approach. Journal of Productivity Analysis 24(1), 93–121). This defines conditional efficiency measures where the production set in the input ×$\times$ output space may depend on the value of the external variables. The statistical properties of nonparametric estimators of these conditional measures are now established (Jeong, S.O., Park, B.U., Simar, L., 2008. Nonparametric conditional efficiency measures: Asymptotic properties. Annals of Operations Research doi: 10.1007/s10479-008-0359-5). These involve the estimation of a nonstandard conditional distribution function which requires the specification of a smoothing parameter (a bandwidth). So far, only the asymptotic optimal order of this bandwidth has been established. This is of little interest for the practitioner. In this paper we fill this gap and we propose a data-driven technique for selecting this parameter in practice. The approach, based on a Least Squares Cross Validation procedure (LSCV), provides an optimal bandwidth that minimizes an appropriate (weighted) integrated Squared Error (ISE). The method is carefully described and exemplified with some simulated data with univariate and multivariate environmental factors. An application on real data (performances of Mutual Funds) illustrates how this new optimal method of bandwidth selection works in practice.     

Efficient simulation and conditional functional limit theorems for ruinous heavy-tailed random walks

The contribution of this paper is to introduce change of measure based techniques for the rare-event analysis of heavy-tailed random walks. Our changes of measures are parameterized by a family of distributions admitting a mixture form. We exploit our methodology to achieve two types of results. First, we construct Monte Carlo estimators that are strongly efficient (i.e. have bounded relative mean squared error as the event of interest becomes rare). These estimators are used to estimate both rare-event probabilities of interest and associated conditional expectations. We emphasize that our techniques allow us to control the expected termination time of the Monte Carlo algorithm even if the conditional expected stopping time (under the original distribution) given the event of interest is infinity–a situation that sometimes occurs in heavy-tailed settings. Second, the mixture family serves as a good Markovian approximation (in total variation) of the conditional distribution of the whole process given the rare event of interest. The convenient form of the mixture family allows us to obtain functional conditional central limit theorems that extend classical results in the literature.     

Multivariate risk measures based on conditional expectation and systemic risk for Exponential Dispersion Models

Exponential dispersion models are well used and studied in quantitative risk management and actuarial science. One of the main interests is the risk measurement analysis of such models when facing extreme loss events. In this paper, we propose two multivariate risk measures based on conditional expectation and derive the explicit formulae for exponential dispersion models. In particular, our multivariate risk measures could facilitate a systemic risk measure with explicit expressions for exponential dispersion models subject to any pre-specified “systemic event.” We provide two numerical examples based on practical data to show the advantages of our approach in the context of exponential dispersion models.     

Asymptotic normality of the size of the giant component via a random walk

In this paper we give a simple new proof of a result of Pittel and Wormald concerning the asymptotic value and (suitably rescaled) limiting distribution of the number of vertices in the giant component of G(n,p) above the scaling window of the phase transition. Nachmias and Peres used martingale arguments to study Karp?s exploration process, obtaining a simple proof of a weak form of this result. We use slightly different martingale arguments to obtain a much sharper result with little extra work.     

Two normality criteria and the converse of the Bloch principle

We prove two normality criteria for a family of meromorphic functions satisfying a certain differential condition and provide a counterexample to the converse of the Bloch principle.     

多维不可微平稳正态过程的极值

讨论了强相关多维不可微平稳正态过程的渐进性质 ,给出了其最大值的极限分布     

Numerical Computation of Multivariate Normal Probabilities

Abstract

The numerical computation of a multivariate normal probability is often a difficult problem. This article describes a transformation that simplifies the problem and places it into a form that allows efficient calculation using standard numerical multiple integration algorithms. Test results are presented that compare implementations of two algorithms that use the transformation with currently available software.     

Two tests for multivariate normality based on the characteristic function

We present two tests for multivariate normality. The presented tests are based on the Lévy characterization of the normal distribution and on the BHEP tests. The tests are affine invariant and consistent. We obtain the asymptotic null distribution of the test statistics using some results about generalized one-sample U-statistics, which are of independent interest.      

Vector-valued multivariate conditional value-at-risk

In this study, we propose a new definition of multivariate conditional value-at-risk (MCVaR) as a set of vectors for arbitrary probability spaces. We explore the properties of the vector-valued MCVaR (VMCVaR) and show the advantages of VMCVaR over the existing definitions particularly for discrete random variables.     

Differential inequalities, normality and quasi-normality

We prove that ifD is a domain in C,α 〉 1 and C 〉 0,then the family F of functions f meromorphic in D such that ｜f′（z）｜/1 ＋ ｜f（z）｜α 〉 C for every z ∈ D is normal in D.For α = 1,the same assumptions imply quasi-normality but not necessarily normality.