Some Truncated Bivariate Distributions

Truncated versions of the bivariate generalized Pareto, bivariate inverted dirichlet and the bivariate Pearson type VII distributions are introduced. Unlike the un-truncated versions, these possess finite moments of all orders and could therefore be better models for certain practical situations. Explicit expressions for the moments are derived for each of the truncated distribution.     

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.     

Distributions of distances and volumes of balls in homogeneous lens spaces

Lens spaces are a family of manifolds that have been a source of many interesting phenomena in topology and differential geometry. Their concrete construction, as quotients of odd-dimensional spheres by a free linear action of a finite cyclic group, allows a deeper analysis of their structure. In this paper, we consider the problem of moments for the distance function between randomly selected pairs of points on homogeneous three-dimensional lens spaces. We give a derivation of a recursion relation for the moments, a formula for the kth moment, and a formula for the moment generating function, as well as an explicit formula for the volume of balls of all radii in these lens spaces.     

Testing Departures from Gamma, Rayleigh and Truncated Normal Distributions

This paper provides necessary and sufficient conditions for a solution to likelihood equations for an exponential family of distributions, which includes Gamma, Rayleigh and singly truncated normal distributions. Furthermore, the maximum likelihood estimator is obtained as a limit case when the equations have no solution. These results provide a way to test departures from Rayleigh and singly truncated normal distributions using the likelihood ratio test. A new easy way to test departures from a Gamma distribution is also introduced.     

MM Algorithms for Some Discrete Multivariate Distributions

The MM (minorization–maximization) principle is a versatile tool for constructing optimization algorithms. Every EM algorithm is an MM algorithm but not vice versa. This article derives MM algorithms for maximum likelihood estimation with discrete multivariate distributions such as the Dirichlet-multinomial and Connor–Mosimann distributions, the Neerchal–Morel distribution, the negative-multinomial distribution, certain distributions on partitions, and zero-truncated and zero-inflated distributions. These MM algorithms increase the likelihood at each iteration and reliably converge to the maximum from well-chosen initial values. Because they involve no matrix inversion, the algorithms are especially pertinent to high-dimensional problems. To illustrate the performance of the MM algorithms, we compare them to Newton’s method on data used to classify handwritten digits.     

二元寿命分布的识别性及参数估计

讨论二元寿命分布的识别性及参数估计,仅是最小值的分布已知时,只有一个参数可识别,当可识最小值的分布已知时,所有参数皆可识别;由此得到了所有参数的最大似然估计.     

Robustness of Tail Index Estimation

Abstract

The implementation of the Hill estimator, which estimates the heaviness of the tail of a distribution, requires a choice of the number of extreme observations in the tails, r from a sample of size n where 2 ≤ r + 1 ≤ n. This article is concerned with a robust procedure of choosing an optimal r. Thus, an estimation procedure, δ _s , based on the idea of spacing statistics, H^(r) is developed. The proposed decision rule for choosing r under the squared error loss is found to be a simple function of the sample size. The proposed rule is then illustrated across a wide range of data, including insurance claims, currency exchange rate returns, and city size.     

Some Classes of Multivariate Life Distributions in Discrete Time

New classes of multivariate survival distribution functions based on monotonic behaviour of a multivariate failure rate are developed in the discrete set up. Relationship among the classes along with multivariate geometric distributions that act as boundaries of the various classes are identified.     

Some Extensions of M-Fractions Related to Strong Stieltjes Distributions

The purpose of this paper is to show the symmetric relations that appear between the coefficients of some even and odd extensions of the M-fractions related to a certain kind of symmetric strong Stieltjes distribution.     

从偏态Pearson VII分布生成的新的多元偏态t分布

一般而言, 偏态的椭球等高分布是一类分布族,有相当一部分的分布都是积分形式, 且此类积分不易求出,而偏态的正态、偏态的正态尺度混合、偏态的PVII型、偏态的PII型的分布却有着很好的结构,偏态t分布属于偏态PVII型分布, 因此,本文在偏态PVII型分布的基础上着重研究新的偏态t分布,给出它的背景、定义、两种随机表示及其等价性.     

Performance Analysis with Truncated Heavy-Tailed Distributions

This paper deals with queues and insurance risk processes where a generic service time, resp. generic claim, has the form U ∧ K for some r.v. U with distribution B which is heavy-tailed, say Pareto or Weibull, and a typically large K, say much larger than . We study the compound Poisson ruin probability ψ(u) or, equivalently, the tail of the M/G/1 steady-state waiting time W. In the first part of the paper, we present numerical values of ψ(u) for different values of K by using the classical Siegmund algorithm as well as a more recent algorithm designed for heavy-tailed claims/service times, and compare the results to different approximations of ψ(u) in order to figure out the threshold between the light-tailed regime and the heavy-tailed regime. In the second part, we investigate the asymptotics as K → ∞ of the asymptotic exponential decay rate γ = γ ^(K) in a more general truncated Lévy process setting, and give a discussion of some of the implications for the approximations. AMS 2000 Subject Classification Primary 68M20, Secondary 60K25 †Partially supported by MaPhySto—A Network in Mathematical Physics and Stochastics, founded by the Danish National Research Foundation. An erratum to this article is available at .     

Some Classes of Life Distributions Characterized by Concave Ordering

1.IntroductionandDefintionsInreliabilitytheory,maintenancetheory,biometricsandeconometrics,variousclassesoflifedistributionsbasedondifferentconceptsofagingareveryuseful.Recently,anumberofclassesoflifedistributionshavebeenintroducedbymeansofdifferentapproachestocharacterizingagingproperty.Oneoftheseapproachesisbasedonthefactthatmanyclassesoflifedistributionmaybecharacterizedbyvariousstochasticorderings,see,e.g.[if.[2],[3]and[4],etc.LetFbethedistributionfunctionofanon-negativerandomvariableXre…     

Continuity Properties of Distributions with Some Decomposability

Absolute continuity and smoothness of distributions in the nested subclasses ~L _m (B), m = 0, 1, 2,..., of the class of all B-decomposable distributions are studied. All invertible matrices are classified into two types in terms of P.V. numbers. The minimum integer m for which all full distributions in ~L _m (B) are absolutely continuous and the minimum integer m for which all absolutely continuous distributions in ~L _m (B) have the densities of class C ^r for 0 r are discussed according to the type of the matrix B related to P.V. numbers.     

多个离散分布的信息差度量的渐近分布

本文中对于多信息源的信息差异性度量B,我们研究了B(u1,u2,…,us)的渐近分布,其中u1,u2,…us是某个概率分布的样本估计．在某些较弱的条件下,我们证明了nB(u1,u2,…,us)的极限分布是自由度为1的X2分布的加权和,其中n是样本容量,u1,u2,…,us不必相互独立．     

Inferences on Correlation Coefficients in Some Classes of Nonnormal Distributions

Correlation coefficients have many applications for studying the relationship among multivariate observations. Classical inferences on correlation coefficients are mainly based on the normality assumption. This assumption is hardly realistic in the real world, which implies that the procedures on correlation coefficients used in many statistical software packages may not be relevant to most data sets in practice. However, we show that the classical procedures, possibly after simple corrections, are also valid in classes of distributions with large skewnesses and heterogeneous marginal kurtoses. A useful class of nonnormal distributions is identified for each of several types of correlation coefficients. The marginals of these distributions may include a variety of univariate distributions with different shapes. The results generalize the classical procedures to much larger classes of distributions than previously known and give a better understanding of the historical controversy regarding the behavior of the sample correlation coefficient. An implication is that one need not be worried so much by the nonnormality of data sets when using these classical procedures, providing simple corrections are evaluated and possibly undertaken.     

Truncated regularized Newton method for convex minimizations

Recently, Li et al. (Comput. Optim. Appl. 26:131–147, 2004) proposed a regularized Newton method for convex minimization problems. The method retains local quadratic convergence property without requirement of the singularity of the Hessian. In this paper, we develop a truncated regularized Newton method and show its global convergence. We also establish a local quadratic convergence theorem for the truncated method under the same conditions as those in Li et al. (Comput. Optim. Appl. 26:131–147, 2004). At last, we test the proposed method through numerical experiments and compare its performance with the regularized Newton method. The results show that the truncated method outperforms the regularized Newton method. The work was supported by the 973 project granted 2004CB719402 and the NSF project of China granted 10471036.     

Marshall and Olkin’s Distributions

A review is provided of the continuous and discrete distributions introduced by the eminent Professors Marshall and Olkin. The topics reviewed include: bivariate geometric distribution, extreme value behavior, bivariate negative binomial distribution, bivariate exponential distribution, concomitants, reliability, distributions of sums and ratios, Ryu’s bivariate exponential distribution, bivariate Pareto distribution and generalized exponential and Weibull distributions. Some hitherto unknown results about these distributions are also mentioned. This is a tribute to the work of Professors Marshall and Olkin.     

On Asymptotic Proximity of Distributions

We consider some general facts concerning the convergence

Multivariate Gaussian Simulation Outside Arbitrary Ellipsoids

Methods for simulation from multivariate Gaussian distributions restricted to be from outside an arbitrary ellipsoidal region are often needed in applications. A standard rejection algorithm that draws a sample from a multivariate Gaussian distribution and accepts it if it is outside the ellipsoid is often employed; however, this is computationally inefficient if the probability of that ellipsoid under the multivariate normal distribution is substantial. We provide a two-stage rejection sampling scheme for drawing samples from such a truncated distribution. Experiments show that the added complexity of the two-stage approach results in the standard algorithm being more efficient for small ellipsoids (i.e., with small rejection probability). However, as the size of the ellipsoid increases, the efficiency of the two-stage approach relative to the standard algorithm increases indefinitely. The relative efficiency also increases as the number of dimensions increases, as the centers of the ellipsoid and the multivariate Gaussian distribution come closer, and as the shape of the ellipsoid becomes more spherical. We provide results of simulation experiments conducted to quantify the relative efficiency over a range of parameter settings.