On two families of bivariate distributions with exponential marginals: Aggregation and capital allocation

In this paper, we consider two main families of bivariate distributions with exponential marginals for a couple of random variables $(X_1,X_2)$. More specifically, we derive closed-form expressions for the distribution of the sum $S=X_1+X_2$, the TVaR of $S$ and the contributions of each risk under the TVaR-based allocation rule. The first family considered is a subset of the class of bivariate combinations of exponentials, more precisely, bivariate combinations of exponentials with exponential marginals. We show that several well-known bivariate exponential distributions are special cases of this family. The second family we investigate is a subset of the class of bivariate mixed Erlang distributions, namely bivariate mixed Erlang distributions with exponential marginals. For this second class of distributions, we propose a method based on the compound geometric representation of the exponential distribution to construct bivariate mixed Erlang distributions with exponential marginals. Notably, we show that this method not only leads to Moran–Downton’s bivariate exponential distribution, but also to a generalization of this bivariate distribution. Moreover, we also propose a method to construct bivariate mixed Erlang distributions with exponential marginals from any absolutely continuous bivariate distributions with exponential marginals. Inspired from Lee and Lin (2012), we show that the resulting bivariate distribution approximates the initial bivariate distribution and we highlight the advantages of such an approximation.     

Recurrence relations for single and product moments of progressive Type-II right censored order statistics from exponential and truncated exponential distributions

In this paper, we establish several recurrence relations satisfied by the single and product moments of progressive Type-II right censored order statistics from an exponential distribution. These relations may then be used, for example, to compute all the means, variances and covariances of exponential progressive Type-II right censored order statistics for all sample sizes n and all censoring schemes (R ₁, R ₂, ..., R _m ), mn. The results presented in the paper generalize the results given by Joshi (1978, Sankhy Ser. B, 39, 362–371; 1982, J. Statist. Plann. Inference, 6, 13–16) for the single moments and product moments of order statistics from the exponential distribution.To further generalize these results, we consider also the right truncated exponential distribution. Recurrence relations for the single and product moments are established for progressive Type-II right censored order statistics from the right truncated exponential distribution.     

Fisher Information in an Order Statistic and its Concomitant

Let (X, Y) have an absolutely continuous distribution with parameter . We suggest regularity conditions on the parent distribution that permit the definition of Fisher information (FI) about in an X-order statistic and its Y-concomitant that are obtained from a random sample from (X, Y). We describe some general properties of the FI in such individual pairs. For the Farlie-Gumbel-Morgenstern parent with dependence parameter , we investigate the properties of this FI, and obtain the asymptotic relative efficiency of the maximum likelihood estimator of for Type II censored bivariate samples. Assuming (X, Y) is Gumbel bivariate exponential of second type, and is the mean of Y, we evaluate the FI in the Y-concomitant of an X-order statistic and compare it with the FI in a single Y-order statistic.     

A characterization of discrete unimodality with applications to variance upper bounds

Bertin and Theodorescu (1984,Statist. Probab. Lett.,2, 23–30) developed a characterization of discrete unimodality based on convexity properties of a discretization of distribution functions. We offer a new characterization of discrete unimodality based on convexity properties of a piecewise linear extension of distribution functions. This reliance on functional convexity, as in Khintchine's classic definition, leads to variance dilations and upper bounds on variance for a large class of discrete unimodal distributions. These bounds are compared to existing inequalities due to Muilwijk (1966,Sankhy, Ser. B,28, p. 183), Moors and Muilwijk (1971,Sankhy, Ser. B,33, 385–388), and Rayner (1975,Sankhy, Ser. B,37, 135–138), and are found to be generally tighter, thus illustrating the power of unimodality assumptions.     

A bivariate exponential distribution arising in random geometry

Summary In this paper a new bivariate exponential distribution, arising naturally in the theory of Poisson line processes, is studied. The distribution has some interesting and useful properties which renders it suitable for use in statistical modelling work. It is presented in the spirit of adding to the repertoire of bivariate exponential forms. It joins other models, such as those of Downton (1970,J. R. Statist. Soc., B,32, 408–417), Marshall and Olkin (1967,J. Appl. Prob.,4, 291–302) and Nagao and Kadoya (1971,Bulletin of the Disaster Prevention Research Institute,20, 3, 183–215), which have their origins in the theory of stochastic processes.     

Finite additivity and upper and lower distribution functions

Summary Suppose U is a set and F is a field of subsets of U. Suppose is a real-nonnegative-valued finitely additive function defined on F and is a function from F into a collection of subsets of .An extension of the «usual» notion of distribution function is introduced by defining the notions of upper and lower distribution functions of with respect to . Two fundamental inequalities, for the case of with vounded range union, are shown. These are shown to imply characterization theorems that imply previous integrability and summability theorems (Ann. di Mat. pur e applic., vol. 96, 1973, pp. 265–287)of the author. A measurability (Boll. U.M.I., (4),vol. 7, 1973, pp. 42–49)characterization theorem is shown; this theorem is a consequence of the above mentioned characterization for -integrability, and in turn implies the -summability characterisation. Set function analogues of the «standard» distribution function integral representation theorems are shown.     

Further developments on some dependence orderings for continuous bivariate distributions

The dependence orderings, more associated and more regression dependent, due to Schriever (1986, Order Dependence, Centre for Mathematics and Computer Sciences, Amsterdam; 1987, Ann. Statist., 15, 1208–1214) and Yanagimoto and Okamoto (1969, Ann. Inst. Statist. Math., 21, 489–505) respectively, are studied in detail for continuous bivariate distributions. Equivalent forms of the orderings under some conditions are given so that the orderings are more easily checkable for some bivariate distributions. For several parametric bivariate families, the dependence orderings are shown to be equivalent to an ordering of the parameter. A study of functionals that are increasing with respect to the more associated ordering leads to inequalities, measures of dependence as well as a way of checking that this ordering does not hold for two distributions.This research has been supported by NSERC Canada grants and a Scientific Grant of the University of Science and Technology of China.     

A characterization of probability distributions by mean values

Let (X) be a measurable complex function onR;X, Y, Z be i.i.d. random variables; and (t, u, v)=E(tX+uY+nZ), wheret, u, vR. In this paper we describe a class of function (x) such that the distribution ofX, Y, Z is determined by the funetion (t, u, v). The main result is a generalization of the author's characterization of normal and stable distributions.     

Exponential trichotomy and p-admissibility for evolution families on the real line

The aim of this paper is to obtain necessary and sufficient conditions for uniform exponential trichotomy of evolution families on the real line. We prove that if p ∈ (1,∞) and the pair (C_b(R,X),C_c(R,X)) is uniformly p-admissible for an evolution family ={U(t,s)}_t≥_s then is uniformly exponentially trichotomic. After that we analyze when the uniform p-admissibility of the pair (C_b(R, X), C_c(R, X)) becomes a necessary condition for uniform exponential trichotomy. As applications of these results we study the uniform exponential dichotomy of evolution families. We obtain that in certain conditions, the admissibility of the pair (C_b(R,X),L^p(R,X)) for an evolution family ={U(t,s)}_t≥_s is equivalent with its uniform exponential dichotomy.     

Local dependence functions for some families of bivariate distributions and total positivity

The purpose of this paper is to investigate a very useful application of a certain local dependence function γ_f(x,y), which was considered recently by Holland and Wang [20]. An interesting property of γ_f(x,y) is that the underlying joint density f(x,y) is TP₂ (that is, totally positive of order 2) if and only if . This gives an elegant way to investigate the TP₂ property of any bivariate distribution. For the Saramanov family, the Ali-Mikhail-Haq family of bivariate distributions and the family of bivariate elliptical distributions, we derive the local dependence function and obtain conditions for f(x,y) to be TP₂. These families are quite rich and include many other large classes of bivariate distributions as their special cases. Similar conditions are obtained for bivariate distributions with exponential conditionals and bivariate distributions with Pareto conditionals.     

Sequential estimation of a parameter of an exponential distribution

We consider the problem of minimum risk point estimation for the parameter =a+b of the exponential distribution with unknown location parameter and scale parameter when the loss function is squared error plus linear cost. In this paper, we propose a sequential estimator of and show that the associated risk is asymptotically one cost less than that given by Ghosh and Mukhopadhyay (1989,South African Statist. J.,23, 251–268).     

A generalization of the results of Pillai

In a recent article Pillai (1990,Ann. Inst. Statist. Math.,42, 157–161) showed that the distribution 1–E (–x ), 0<1; 0x, whereE (x) is the Mittag-Leffler function, is infinitely divisible and geometrically infinitely divisible. He also clarified the relation between this distribution and a stable distribution. In the present paper, we generalize his results by using Bernstein functions. In statistics, this generalization is important, because it gives a new characterization of geometrically infinitely divisible distributions with support in (0, ).     

A conglomerate of exponential supercategories of the category of finitely generated topological spaces

Summary For every ordinaln > 1 we define a categoryT _n of topological spaces in ech's sense which is isomorphic to a category ofn-ary monorelational systems. We show that every categoryT _n is an exponential supercategory of the categoryB of finitely generated topological spaces, which means that well-behaved function spacesG ^H can be defined inT _n wheneverG B.     

Behaviour of Exponential Splines as Tensions Increase without Bound

Schweikert (J. Math. Phys.45(1966), 312–317) showed that for sufficiently high tensions an exponential spline would have no more changes in sign of its second derivative than there were changes in the sign of successive second differences of its knot sequence. Späth (Computing4(1969), 225–233) proved the analogous result for first derivatives, assuming uniform tension throughout the spline. Later, Pruess (J. Approx. Theory17(1976), 86–96) extended Späth's result to the case where the inter-knot tensionsp_imay not all be the same but tend to infinity at the same asymptotic growth rate, in the sense thatp_i∈Θ(p₁) for alli. This paper extends Pruess's result by showing his hypothesis of uniform boundedness of the tensions to be unnecessary. A corollary is the fact that for high enough minimum interknot tension, the exponential spline through monotone knots will be a ²monotone curve. In addition, qualitative bounds on the difference in slopes between the interpolating polygon and the exponential spline are developed, which show that Gibbs-like behaviour of the spline's derivative cannot occur in the neighbourhood of the knots.     

The expected values of invariant polynomials with matrix argument of elliptical distributions

ThisresearchissupportedbytheChineseAcademyofSciences.1.IntroductionTherearemailypublicatiollsonmatrixvariatedistributions(ormatrixdistributionforsimplicity),inparticular,abollttheirexpectedvaluesofzonalpolynomialsoftheirquadraticforms(of.[12],[151,[171and…     

二元Block～Basu型指数分布的特征及其应用

导出了二元Block~Basu型指数分布的一个特征,利用该特征,获得了二元Block~Basu型指数分布参数的最大似然估计及矩估计,给出了强度服从二元Block~Basu型分布时并联结构系统可靠度的估计,并给出了二元Block~Basu型指数分布的一个随机模拟.     

Decomposition of Exponential Distributions on Positive Semigroups

Let (S,·) be a positive semigroup and T a sub-semigroup of S. In many natural cases, an element can be factored uniquely as x=yz, where and where z is in an associated “quotient space” S/T. If X has an exponential distribution on S, we show that Y and Z are independent and that Y has an exponential distribution on T. We prove a converse when the sub-semigroup is for . Specifically, we show that if Y _t and Z _t are independent and Y _t has an exponential distribution on S _t for each , then X has an exponential distribution on S. When applied to ([0,∞), +) and , these results unify and extend known results on the quotient and remainder when X is divided by t.     

Monotone path systems in simple regions

A monotone path system (MPS) is a finite set of pairwise disjoint paths (polygonal areas) in thexy-plane such that every horizontal line intersects each of the paths in at most one point. A MPS naturally determines a pairing of its top points with its bottom points. We consider a simple polygon in thexy-plane wich bounds the simple polygonal (closed) regionD. LetT andB be two finite, disjoint, equicardinal sets of points ofD. We give a good characterization for the existence of a MPS inD which pairsT withB, and a good algorithm for finding such a MPS, and we solve the problem of finding all MPSs inD which pairT withB. We also give sufficient conditions for any such pairing to be the same.The first author's research is supported by the Natural Sciences and Engineering Research Council of Canada