Characterizations of mixtures of discrete distributions by a regression point

Given that the conditional distribution p_s(y|x) of Y, given X = x is an x-fold convolution of a nonnegative integer-valued r.v. ξ for every s= P[ξ = 0] > 0, the distribution of X, hence also of Y, is characterized by the regression point m(0) = E[X|Y = 0]. An infinite variety of generalized distributions (of Y) can be characterized by arbitrarily varying the distribution of X.     

Characterizations of discrete distributions by a conditional distribution and a regression function

Summary The bivariate distribution of (X, Y), whereX andY are non-negative integer-valued random variables, is characterized by the conditional distribution ofY givenX=x and a consistent regression function ofX onY. This is achieved when the conditional distribution is one of the distributions: a) binomial, Poisson, Pascal or b) a right translation of these. In a) the conditional distribution ofY is anx-fold convolution of another random variable independent ofX so thatY is a generalized distribution. A main feature of these characterizations is that their proof does not depent on the specific form of the regression function. It is also indicated how these results can be used for good-ness-of-fit purposes.     

Characterizations of multivariate life distributions

Characterizations of multivariate distributions has been a topic of great interest in applied statistics literature for the last three decades. In this paper, we develop characterizations of multivariate lifetime distributions by relationship between multivariate failure rates (reversed failure rates) and the left (right) truncated expectations of functions of random variables. We, then, discuss the application of the results to derive a multivariate Stein type identity.     

On strong unimodality of multivariate discrete distributions

A discrete function f defined on Zⁿ is said to be logconcave if for , , . A more restrictive notion is strong unimodality. Following Barndorff-Nielsen [O. Barndorff-Nielsen, Unimodality and exponential families, Commun. Statist. 1 (1973) 189-216] a discrete function is called strongly unimodal if there exists a convex function such that  if . In this paper sufficient conditions that ensure the strong unimodality of a multivariate discrete distribution, are given. Examples of strongly unimodal multivariate discrete distributions are presented.     

Simulation of certain multivariate generalized Pareto distributions

The investigation of multivariate generalized Pareto distributions (GPDs) has begun only recently. For further progress with these distributions simulation methods are an important part. We describe several methods of simulating GPDs, beginning with an efficient method for the logistic GPD. The algorithm is based on the Shi transformation, which was already used for the simulation of multivariate extreme value distributions (EVDs) of logistic type. In the sequel another algorithm is presented simulating a broader class of GPDs. Due to its numerical complexity it is only practicably applicable in low dimensions. A method is given to generate unconditional GPD random vectors from conditionally GPD distributed random vectors. A short application of the simulation methods in the analysis of a real hydrological data set concludes the article. The simulation algorithms are available on the author’s home page .      

Peaks-over-threshold stability of multivariate generalized Pareto distributions

It is well-known that the univariate generalized Pareto distributions (GPD) are characterized by their peaks-over-threshold (POT) stability. We extend this result to multivariate GPDs.It is also shown that this POT stability is asymptotically shared by distributions which are in a certain neighborhood of a multivariate GPD. A multivariate extreme value distribution is a typical example.The usefulness of the results is demonstrated by various applications. We immediately obtain, for example, that the excess distribution of a linear portfolio with positive weights a_i, i≤d, is independent of the weights, if (U₁,…,U_d) follows a multivariate GPD with identical univariate polynomial or Pareto margins, which was established by Macke [On the distribution of linear combinations of multivariate EVD and GPD distributed random vectors with an application to the expected shortfall of portfolios, Diploma Thesis, University of Würzburg, 2004, (in German)] and Falk and Michel [Testing for tail independence in extreme value models. Ann. Inst. Statist. Math. 58 (2006) 261-290]. This implies, for instance, that the expected shortfall as a measure of risk fails in this case.     

Some notes on multivariate generalized Pareto distributions

The investigation of multivariate generalized Pareto distributions (GPDs) has begun only recently and there are slightly varying definitions of GPDs available. In this article we investigate the one from Section 5.1 of Falk et al. [Laws of Small Numbers: Extremes and Rare Events, second ed., Birkhäuser, Basel, 2004], which does not differ in the area of interest from those of other authors. We first give an interpretation of the case of independence in terms of the peaks-over-threshold approach. This case is also used in dimension d=3 by Falk et al. [Laws of Small Numbers: Extremes and Rare Events, second ed., Birkhäuser, Basel, 2004] as a counterexample to show that GP functions are not necessarily distribution functions on their entire support. We generalize this counterexample to an arbitrary dimension d≥3 and demonstrate also that other GP functions show this behavior. Finally we show that different GPDs can lead to the same conditional probability measure in the area of interest.     

Canonical representation of conditionally specified multivariate discrete distributions

Most work on conditionally specified distributions has focused on approaches that operate on the probability space, and the constraints on the probability space often make the study of their properties challenging. We propose decomposing both the joint and conditional discrete distributions into characterizing sets of canonical interactions, and we prove that certain interactions of a joint distribution are shared with its conditional distributions. This invariance opens the door for checking the compatibility between conditional distributions involving the same set of variables. We formulate necessary and sufficient conditions for the existence and uniqueness of discrete conditional models, and we show how a joint distribution can be easily computed from the pool of interactions collected from the conditional distributions. Hence, the methods can be used to calculate the exact distribution of a Gibbs sampler. Furthermore, issues such as how near compatibility can be reconciled are also discussed. Using mixed parametrization, we show that the proposed approach is based on the canonical parameters, while the conventional approaches are based on the mean parameters. Our advantage is partly due to the invariance that holds only for the canonical parameters.     

Family of multivariate generalized t distributions

In this paper, we introduce a new family of multivariate distributions as the scale mixture of the multivariate power exponential distribution introduced by Gómez et al. (Comm. Statist. Theory Methods 27(3) (1998) 589) and the inverse generalized gamma distribution. Since the resulting family includes the multivariate t distribution and the multivariate generalization of the univariate GT distribution introduced by McDonald and Newey (Econometric Theory 18 (11) (1988) 4039) we call this family as the “multivariate generalized t-distributions family”, or MGT for short. We show that this family of distributions belongs to the elliptically contoured distributions family, and investigate the properties. We give the stochastic representation of a random variable distributed as a multivariate generalized t distribution. We give the marginal distribution, the conditional distribution and the distribution of the quadratic forms. We also investigate the other properties, such as, asymmetry, kurtosis and the characteristic function.     

Characterizations of generalized quadrangles by generalized homologies

Let S = (P, B, I) be a generalized quadrangle of order (s, t). For x, y P, x y, let (x, y) be the group of all collineations of S fixing x and y linewise. If z {x, y}, then the set of all points incident with the line xz (resp. yz) is denoted by (resp. ). The generalized quadrangle S = (P, B, I) is said to be (x, y)-transitive, x y, if (x, y) is transitive on each set and . If S = (P, B, I) is a generalized quadrangle of order (s, t), s > 1 and t > 1, which is (x, y)-transitive for all x, y P with x y, then it is proved that we have one of the following: (i) S W(s), (ii) S Q(4, s), (iii) S H(4, s), (iv) S Q(5, s), (v) s = t² and all points are regular.     

Characterizations and closure under convolution of two classes of multivariate life distributions

Let ?(η) be the class of positive random vectors $\text{T}$ for which min_1?i?nα_iT_i is IFRA (NBU) for all α_i > 0, i = 1,…,n where n is an arbitrary positive integer. Characterizations of the classes ? and η are obtained and utilized to show that η is closed under convolution and that ? is closed under convolution provided one of the two convoluted vectors has independent components.     

Some properties and characterizations for generalized multivariate Pareto distributions

In this paper, several distributional properties and characterization theorems of the generalized multivariate Pareto distributions are studied. It is found that the multivariate Pareto distributions have many mixture properties. They are mixed either by geometric, Weibull, or exponential variables. The multivariate Pareto, MP^(k)(I), MP^(k)(II), and MP^(k)(IV) families have closure property under finite sample minima. The MP^(k)(III) family is closed under both geometric minima and geometric maxima. Through the geometric minima procedure, one characterization theorem for MP^(k)(III) distribution is developed. Moreover, the MP^(k)(III) distribution is proved as the limit multivariate distribution under repeated geometric minimization. Also, a characterization theorem for the homogeneous MP^(k)(IV) distribution via the weighted minima among the ordered coordinates is developed. Finally, the MP^(k)(II) family is shown to have the truncation invariant property.     

On a new system of discrete distributions and characterizations of several discrete distributions by equality of distributions

This paper deals with a new system of discrete distributions. It also gives several characterizations of the Waring (and hence the Yule) distribution (and its truncated versions), the super-Poisson, the discrete uniform and other discrete distributions by using this system and other such systems existing in the literature, and linear regression. Continuous analogues of the above results are also briefly discussed.     

Multiparameter estimation for some multivariate discrete distributions with possibly dependent components

Summary In multiparameter estimation for multivariate discrete distributions with infinite support, inadmissibility problems in situations where the multivariate probability distribution function isnot a product of the one-dimensional marginal probability distribution functions have previously been unexplored. This paper examines the inadmissibility problem in some of these situations. Special attention is given to estimating the mean of a negative multinomial distribution. In estimating the mean vector, certain Clevenson-Zidek type estimators are shown to be uniformly better than the usual estimator under a large class of generally scaled squared loss functions. Some of the results are generalized to other multivariate discrete distributions and to situations where several independent negative multinomial distributions are considered.     

A generalized Pólya urn model and related multivariate distributions

In this paper, we study of Pólya urn model containing balls of (m+1) different labels under a general replacement scheme, which is characterized by an (m+1) × (m+1) addition matrix of integers without constraints on the values of these (m+1)² integers other than non-negativity. LetX ₁,X ₂,...,X _n be trials obtained by the Pólya urn scheme (with possible outcomes: “O”, “1”,...“m”). We consider the multivariate distributions of the numbers of occurrences of runs of different types arising from the various enumeration schemes and give a recursive formula of the probability generating function. Some closed form expressions are derived as special cases, which have potential applications to various areas. Our methods for the derivation of the multivariate run-related distribution are very simple and suitable for numerical and symbolic calculations by means of computer algebra systems. The results presented here develop a general workable framework for the study of Pólya urn models. Our attempts are very useful for understanding non-classic urn models. Finally, numerical examples are also given in order to illustrate the feasibility of our results. This research was partially supported by the ISM Cooperative Research Program (2003-ISM·CRP-2007).     

Characterizations of mixture of two-component exponentiated family of distributions based on generalized order statistics

In this paper, we shall characterize mixture of two components of exponentiated family of distributions based on recurrence relations for moment and conditional moment generating functions of generalized order statistics. Results for ordinary order statistics and upper kth record values are obtained as special cases.