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.     

On first passage time structure of random walks

For continuous time birth-death processes on {0,1,2,…}, the first passage time T⁺_n from n to n + 1 is always a mixture of (n + 1) independent exponential random variables. Furthermore, the first passage time T_0,n+1 from 0 to (n + 1) is always a sum of (n + 1) independent exponential random variables. The discrete time analogue, however, does not necessarily hold in spite of structural similarities. In this paper, some necessary and sufficient conditions are established under which T⁺_n and T_0,n+1 for discrete time birth-death chains become a mixture and a sum, respectively, of (n + 1) independent geometric random variables on {1,2,…};. The results are further extended to conditional first passage times.     

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.     

On the unimodality of power transformations of positive stable densities

Let Z_α be a positive α‐stable random variable and $r\in {\mathbb {R}}.Let Z_α be a positive α‐stable random variable and $r\in {\mathbb {R}}.$ We show the existence of an unbounded open domain D in [1/2, 1] × ( ? ∞, ?1/2] with a cusp at (1/2, ?1/2), characterized by the complete monotonicity of the function $F_{\alpha ,r}(\lambda ) = (\alpha \lambda ^\alpha -r)e^{-\lambda ^\alpha }\!\! ,$ such that $Z_\alpha ^r$ is unimodal if and only if (α, r)?D.     

Independent set and matching permutations

Let $G$ be a graph $G$ whose largest independent set has size $m$. A permutation $\pi$ of $\left\{1,\text{…},m\right\}$ is an independent set permutation of $G$ if $$a_{\pi \left(1\right)}\left(G\right)\le a_{\pi \left(2\right)}\left(G\right)\le \mathrm{?}\le a_{\pi \left(m\right)}\left(G\right),$$ where $a_k\left(G\right)$ is the number of independent sets of size $k$ in $G$. In 1987 Alavi, Malde, Schwenk, and Erd?s proved that every permutation of $\left\{1,\text{…},m\right\}$ is an independent set permutation of some graph with $\alpha \left(G\right)=m$, that is, with the largest independent set having size $m$. They raised the question of determining, for each $m$, the smallest number $f\left(m\right)$ such that every permutation of $\left\{1,\text{…},m\right\}$ is an independent set permutation of some graph with $\alpha \left(G\right)=m$ and with at most $f\left(m\right)$ vertices, and they gave an upper bound on $f\left(m\right)$ of roughly $m^{2m}$. Here we settle the question, determining $f\left(m\right)=m^m$, and make progress on a related question, that of determining the smallest order such that every permutation of $\left\{1,\text{…},m\right\}$ is the unique independent set permutation of some graph of at most that order. More generally we consider an extension of independent set permutations to weak orders, and extend Alavi et al.'s main result to show that every weak order on $\left\{1,\text{…},m\right\}$ can be realized by the independent set sequence of some graph with $\alpha \left(G\right)=m$ and with at most $m^{m+2}$ vertices. Alavi et al. also considered matching permutations, defined analogously to independent set permutations. They observed that not every permutation of $\left\{1,\text{…},m\right\}$ is a matching permutation of some graph with the largest matching having size $m$, putting an upper bound of $2^{m?1}$ on the number of matching permutations of $\left\{1,\text{…},m\right\}$. Confirming their speculation that this upper bound is not tight, we improve it to $O\left(2^m∕\sqrt{m}\right)$.     

Unimodal Gorenstein h-vectors without the Stanley–Iarrobino property

The study of the h-vectors of graded Gorenstein algebras is an important topic in combinatorial commutative algebra, which despite the large amount of literature produced during the last several years, still presents many interesting open questions. In this note, we commence a study of those unimodal Gorenstein h-vectors that do not satisfy the Stanley–Iarrobino property. Our main results, which are characteristic free, show that such h-vectors exist: 1) In socle degree e if and only if e≥6; and 2) in every codimension five or greater. The main case that remains open is that of codimension four, where no Gorenstein h-vector is known without the Stanley–Iarrobino property. We conclude by proposing the following very general conjecture: The existence of any arbitrary level h-vector is independent of the characteristic of the base field.     

Weak unimodality of finite measures, and an application to potential theory of additive Lévy processes

A probability measure on is called weakly unimodal if there exists a constant such that for all 0$">,

(0.1)

Here, denotes the -ball centered at with radius 0$">.

In this note, we derive a sufficient condition for weak unimodality of a measure on the Borel subsets of . In particular, we use this to prove that every symmetric infinitely divisible distribution is weakly unimodal. This result is then applied to improve some recent results of the authors on capacities and level sets of additive Lévy processes.

     

Modelling the process of incoming problem reports on released software products

For big software developing companies, it is important to know the amount of problems of a new software product that are expected to be reported in a period after the date of release, on a weekly basis. For each of a number of past releases, weekly data are present on the number of such reports. Based on the type of data that is present, we construct a stochastic model for the weekly number of problems to be reported. The (non‐parametric) maximum likelihood estimator for the crucial model parameter, the intensity of an inhomogeneous Poisson process, is defined. Moreover, the expectation maximization algorithm is described, which can be used to compute this estimate. The method is illustrated using simulated data. Copyright © 2004 John Wiley & Sons, Ltd.     

Distributions of the Numbers of Failures and Successes in a Waiting Time Problem

Abstract. In this article we consider infinite sequences of Bernoulli trials and study the exact and asymptotic distribution of the number of failures and the number of successes observed before the r-th appearance of a pair of successes separated by a pre-specified number of failures. Several formulae are provided for the probability mass function, probability generating function and moments of the distribution along with some asymptotic results and a Poisson limit theorem. A number of interesting applications in various areas of applied science are also discussed.     

Influence of the prior distribution on the risk of the Bayes rule

The risk influence function is defined as the directional derivative of the risk of the Bayes rule. The properties of this function are studied and the relationship between unimodal prior distribution and the shape of the frequentist risk of the corresponding Bayes procedure is examined.