A note on runs of geometrically distributed random variables

Recently, Grabner et al. [Combinatorics of geometrically distributed random variables: run statistics, Theoret. Comput. Sci. 297 (2003) 261-270] and Louchard and Prodinger [Ascending runs of sequences of geometrically distributed random variables: a probabilistic analysis, Theoret. Comput. Sci. 304 (2003) 59-86] considered the run statistics of geometrically distributed independent random variables. They investigated the asymptotic properties of the number of runs and the longest run using the corresponding probability generating functions and a Markov chain approach. In this note, we reconsider the asymptotic properties of such statistics using another approach. Our approach of finding the asymptotic distributions is based on the construction of runs in a sequence of m-dependent random variables. This approach enables us to find the asymptotic distributions of many run statistics via the theorems established for m-dependent sequence of random variables. We also provide the asymptotic distribution of the total number of non-decreasing runs and the longest non-decreasing run.     

The generating function of Whitworth runs

Let G be a graph. The number of ways of selecting k vertices in G such that the subgraph induced by the k selected vertices containing l edges may be considered as Whitworth runs. For two arbitrary graphs G₁ and G₂ we show that the generating function of G₁ can be written as a sum of the generating function of G₂. As an application we derive a difference equation satisfied by the generating function of a line graph and that of a cycle graph. Two independent solutions in the closed-form are found. One is equivalent to the Whitworth bracelet problem with two colors. Furthermore, a line and a cycle graph and a two-line graph have been studied. We show that all solutions can be written as a sum of the solutions for single-line cases.     

Conditional and unconditional distributions forM/G/1 type queues with server vacations

M/G/1 queues with server vacations have been studied extensively over the last two decades. Recent surveys by Boxma [3], Doshi [5] and Teghem [14] provide extensive summary of literature on this subject. More recently, Shanthikumar [11] has generalized some of the results toM/G/1 type queues in which the arrival pattern during the vacations may be different from that during the time the server is actually working. In particular, the queue length at the departure epoch is shown to decompose into two independent random variables, one of which is the queue length at the departure epoch (arrival epoch, steady state) in the correspondingM/G/1 queue without vacations. Such generalizations are important in the analysis of situations involving reneging, balking and finite buffer cyclic server queues. In this paper we consider models similar to the one in Shanthikumar [11] but use the work in the system as the starting point of our investigation. We analyze the busy and idle periods separately and get conditional distributions of work in the system, queue length and, in some cases, waiting time. We then remove the conditioning to get the steady state distributions. Besides deriving the new steady state results and conditional waiting time and queue length distributions, we demonstrate that the results of Boxma and Groenendijk [2] follow as special cases. We also provide an alternative approach to deriving Shanthikumar's [11] results for queue length at departure epochs.     

Relative ageing for two parallel systems and related problems

Kalashnikov and Rachev [1] have proposed a partial ordering of two life distributions which is equivalent to an increasing hazard (failure rate) ratio, when the ratio exists. The phenomenon of crossing hazards has received considerable attention in recent years. Recently, Sengupta and Deshpande [2) have studied this and two other models of relative ageing. In this paper, we consider the relative ageing properties of two parallel systems with identical but different number of components. We also compare the variances of the two life distributions having the same mean but with increasing hazard ratio. Several examples are given to illustrate the results.     

相依二值随机变量序列游程的一类强极限定理

本文利用文献[1]与[2]中提出的方法和思想,研究相依二值随机变量序列游程的极限性质,得到游程平均数的一类强极限定理.     

Shock models and the MIFRA property

Marshall and Shaked [6] have shown that some multivariate life distributions obtained from their shock model satisfy the IFRA conditions A and B of Esary and Marshall [5]. Block and Savits [2] have introduced a multivariate IFRA condition which is stronger than Conditions A and B. In this paper it is shown that the multivariate life distributions of Marshall and Shaked actually satisfy the Block-Savits MIFRA condition. As a consequence it follows that the damage processes associated with the Marshall-Shaked shock models are multivariate strongly IFRA in the sense of Block and Savits [3].     

On Riesz and Wishart distributions associated with decomposable undirected graphs

Classical Wishart distributions on the open convex cone of positive definite matrices and their fundamental features are extended to generalized Riesz and Wishart distributions associated with decomposable undirected graphs using the basic theory of exponential families. The families of these distributions are parameterized by their expectations/natural parameter and multivariate shape parameter and have a non-trivial overlap with the generalized Wishart distributions defined in Andersson and Wojnar (2004) [4] and [8]. This work also extends the Wishart distributions of type I in Letac and Massam (2007) [7] and, more importantly, presents an alternative point of view on the latter paper.     

Moment properties of multivariate infinitely divisible laws and criteria for multivariate self-decomposability

Ramachandran (1969) [9, Theorem 8] has shown that for any univariate infinitely divisible distribution and any positive real number α, an absolute moment of order α relative to the distribution exists (as a finite number) if and only if this is so for a certain truncated version of the corresponding Lévy measure. A generalized version of this result in the case of multivariate infinitely divisible distributions, involving the concept of g-moments, was given by Sato (1999) [6, Theorem 25.3]. We extend Ramachandran’s theorem to the multivariate case, keeping in mind the immediate requirements under appropriate assumptions of cumulant studies of the distributions referred to; the format of Sato’s theorem just referred to obviously varies from ours and seems to have a different agenda. Also, appealing to a further criterion based on the Lévy measure, we identify in a certain class of multivariate infinitely divisible distributions the distributions that are self-decomposable; this throws new light on structural aspects of certain multivariate distributions such as the multivariate generalized hyperbolic distributions studied by Barndorff-Nielsen (1977) [12] and others. Various points relevant to the study are also addressed through specific examples.     

Joint distributions of numbers of success runs of specified lengths in linear and circular sequences

In this paper, we study two joint distributions of the numbers of success runs of several lengths in a sequence ofn Bernoulli trials arranged on a line (linear sequence) or on a circle (circular sequence) based on four different enumeration schemes. We present formulae for the evaluation of the joint probability functions, the joint probability generating functions and the higher order moments of these distributions. Besides, the present work throws light on the relation between the joint distributions of the numbers of success runs in the circular and linear binomial model. We give further insights into the run-related problems arisen from the circular sequence. Some examples are given in order to illustrate our theoretical results. Our results have potential applications to other problems such as statistical run tests for randomness and reliability theory. This research was partially supported by the ISM Cooperative Research Program (2003-ISM.CRP-2007).     

奖惩系统在随机优序下的极小与极大平稳分布

奖惩系统(Bonus-M a lus System)是世界各国机动车辆险中广泛采用的一种经验费率厘定机制.文[1]在最一般的框架下,给出了奖惩系统的数学建模与稳态分析.本文将进一步证明,文[1]中给出的奖惩系统两种特定的平稳分布恰分别是奖惩系统在随机优序下的极小与极大平稳分布.特别地,基于这一事实严格证明了文[1]提出的有关封闭型奖惩系统年度总保费的一个猜想.     

Tangent measure distributions of hyperbolic Cantor sets

Tangent measure distributions were introduced byBandt [2] andGraf [8] as a means to describe the local geometry of self-similar sets generated by iteration of contractive similitudes. In this paper we study the tangent measure distributions of hyperbolic Cantor sets generated by certain contractive mappings, which are not necessarily similitudes. We show that the tangent measure distributions of these sets equipped with either Hausdorff- or Gibbs measure are unique almost everywhere and give an explicit formula describing them as probability distributions on the set of limit models ofBedford andFisher [5].     

The Quasiasymptotic Expansion at the Origin. Abelian-Type Results for Laplace and Stieltjes Transforms

The notion of quasiasymptotic expansion at the origin of tempered distributions supported by [0, α] and the structural theorem for such distributions that have the quasiasymptotic expansion at the origin are given. As applications, the Abelian-type results for the distributional Stieltjes and Laplace transforms are obtained.     

Probability Distribution of Solution Time in GRASP: An Experimental Investigation

A GRASP (greedy randomized adaptive search procedure) is a multi-start metaheuristic for combinatorial optimization. We study the probability distributions of solution time to a sub-optimal target value in five GRASPs that have appeared in the literature and for which source code is available. The distributions are estimated by running 12,000 independent runs of the heuristic. Standard methodology for graphical analysis is used to compare the empirical and theoretical distributions and estimate the parameters of the distributions. We conclude that the solution time to a sub-optimal target value fits a two-parameter exponential distribution. Hence, it is possible to approximately achieve linear speed-up by implementing GRASP in parallel.     

Multivariate measures of skewness for the skew-normal distribution

The main objective of this work is to calculate and compare different measures of multivariate skewness for the skew-normal family of distributions. For this purpose, we consider the Mardia (1970) [10], Malkovich and Afifi (1973) [9], Isogai (1982) [17], Srivastava (1984) [15], Song (2001) [14], Móri et al. (1993) [11], Balakrishnan et al. (2007) [3] and Kollo (2008) [7] measures of skewness. The exact expressions of all measures of skewness, except for Song’s, are derived for the family of skew-normal distributions, while Song’s measure of shape is approximated by the use of delta method. The behavior of these measures, their similarities and differences, possible interpretations, and their practical use in testing for multivariate normal are studied by evaluating their power in the case of some specific members of the multivariate skew-normal family of distributions.     

A Functional Equation Whose Unknown is \mathcal{P}([0,1]) Valued

We study a functional equation whose unknown maps a Euclidean space into the space of probability distributions on [0,1]. We prove existence and uniqueness of its solution under suitable regularity and boundary conditions, we show that it depends continuously on the boundary datum, and we characterize solutions that are diffuse on [0,1]. A canonical solution is obtained by means of a Randomly Reinforced Urn with different reinforcement distributions having equal means. The general solution to the functional equation defines a new parametric collection of distributions on [0,1] generalizing the Beta family.     

Improved Approximation Algorithms for Tree Alignment

Multiple sequence alignment is a task at the heart of much of current computational biology[4]. Several different objective functions have been proposed to formalize the task of multiple sequence alignment, but efficient algorithms are lacking in each case. Thus multiple sequence alignment is one of the most critical, essentially unsolved problems in computational biology. In this paper we consider one of the more compelling objective functions for multiple sequence alignment, formalized as thetree alignment problem. Previously in[13], a ratio-two approximation method was developed for tree alignment, which ran incubictime (as a function of the number of fixed length strings to be aligned), along with a polynomial time approximation scheme (PTAS) for the problem. However, the PTAS in[13]had a running time which made it impractical to reduce the performance ratio much below two for small size biological sequences (100 characters long). In this paper we first develop a ratio-two approximation algorithm which runs inquadratictime, and then use it to develop a PTAS which has a better performance ratio and a vastly improved worst case running time compared to the scheme in[13]for the case where the given tree is a regular deg-ary tree. With the new approximation scheme, it is now practical to guarantee a ratio of 1.583 for strings of lengths 200 characters or less.     

On generalized order statistics and maximal correlation as a measure of dependence

In the first part of this review article some recent developments of maximal correlation coefficient, introduced by Gebelein (1941) [7], and its applications in various areas of statistics are discussed. The second part is devoted to find the distributions providing the maximal correlation coefficient between generalized order statistics (gos) and dual generalized order statistics (dgos), which are introduced by Kamps (1995) [8] and Burkschat et al. (2003) [4], respectively. Finally, in the third part, general theorems are presented, which give simple non-parametric criterion for the asymptotic independence between the different elements of gos, as well as dgos.     

One Sample Tests for the Location of Modes of Nonnormal Data

There are few techniques available for testing if modes take specified values. We show that standard tests of location such as the t-test and the Wilcoxon test, which test for the mean and median respectively, can perform poorly as tests for modes when the data is other than unimodal and symmetric. Carolan and Rayner [1] proposed a score test of location for symmetric nonnormal data. We consider a family of distributions similar to those considered by Carolan and Rayner [1] and propose a test for the mode or modes of data from multimodal or skewed distributions and demonstrate by way of simulations that it is reasonably effective.     

Cosheaves and distributions on toposes

Distributions on a Grothendieck topos were introduced by Lawvere [12] (cf. also [13]) as a generalization of the classical notion (cf. [20]) of real-valued distributions on a topological space. The cosheaves approach to distributions which is implicit in work of Pitts [19] is used here first, in order to answer affirmatively a question posed in [12] concerning the existence of the symmetric topos and next, in order to prove a structure theorem for categories of distributions on Grothendieck toposes that is similar in spirit to the Joyal-Tierney [11] structure theorem for Grothendieck toposes.Dedicated to the memory of Alan Day.Presented by J. Sichler.