On the minimality of search paths under absolutely continuous univariate distributions

The linear search problem is concerned with the search for a target located randomly on a line. The target being sought for might be on either side of some starting point. A searcher makes for the target with an upper bound on his speed, using a continuous search path in which he changes his direction many times before reaching his goal. The minimality of average distance (time) from the starting point upon reaching the target is the measure of optimality of search paths. We have obtained sufficient conditions that guarantee the minimality of such search paths whenever the target has an absolutely continuous distribution.

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Zusammenfassung Ein Objekt, das sich in zufälliger Position auf einer Strecke befindet, soll gesucht werden; der Startpunkt liegt im Inneren der Strecke. Für die Bewegungsgeschwindigkeit bei der Suche ist eine obere Schranke gegeben; der Suchpfad muß stetig sein, wobei beliebig viele Richtungsänderungen zugelassen sind. Das Ziel ist die Minimierung der erwarteten Zeit bis zum Erreichen des Objekts. Unter der Voraussetzung einer absolut stetigen Verteilung für die Position des Objekts werden hinreichende Bedingungen für die Optimalität eines Suchpfades angegeben.

     

On the moments and distribution of discrete Choquet integrals from continuous distributions

We study the moments and the distribution of the discrete Choquet integral when regarded as a real function of a random sample drawn from a continuous distribution. Since the discrete Choquet integral includes weighted arithmetic means, ordered weighted averaging functions, and lattice polynomial functions as particular cases, our results encompass the corresponding results for these aggregation functions. After detailing the results obtained in [J.-L. Marichal, I. Kojadinovic, Distribution functions of linear combinations of lattice polynomials from the uniform distribution, Statistics & Probability Letters 78 (2008) 985–991] in the uniform case, we present results for the standard exponential case, show how approximations of the moments can be obtained for other continuous distributions such as the standard normal, and elaborate on the asymptotic distribution of the Choquet integral. The results presented in this work can be used to improve the interpretation of discrete Choquet integrals when employed as aggregation functions.     

Geometrically concave univariate distributions

In this paper our aim is to show that if a probability density function is geometrically concave (convex), then the corresponding cumulative distribution function and the survival function are geometrically concave (convex) too, under some assumptions. The proofs are based on the so-called monotone form of l'Hospital's rule and permit us to extend our results to the case of the concavity (convexity) with respect to Hölder means. To illustrate the applications of the main results, we discuss in details the geometrical concavity of the probability density function, cumulative distribution function and survival function of some common continuous univariate distributions. Moreover, at the end of the paper, we present a simple alternative proof to Schweizer's problem related to the Mulholland's generalization of Minkowski's inequality.     

On discrete distributions of orderk

Summary The class of discrete distributions of orderk is defined as the class of the generalized discrete distributions with generalizer a discrete distribution truncated at zero and from the right away fromk+1. The probability function and factorial moments of these distributions are expressed in terms of the (right) truncated Bell (partition) polynomials and several special cases are briefly examined. Finally a Poisson process of orderk, leading in particular to the Poisson distribution of orderk, is discussed.     

On discrete distributions of orderk

Summary This paper gives some results on calculation of probabilities and moments of the discrete distributions of orderk. Further, a new distribution of orderk, which is called the logarithmic series distribution of orderk, is investigated. Finally, we discuss the meaning of theorder of the distributions. The Institute of Statistical Mathematics     

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.     

On discrete and continuous wirtinger inequalities

We shall present several generalizations of discrete Wirtinger's inequality, and establish their continuous analogs.     

On the resolution of certain discrete univariate max–min problems

Graph coloring is a classical NP-hard combinatorial optimization problem with many practical applications. A broad range of heuristic methods exist for tackling the graph coloring problem: from fast greedy algorithms to more time-consuming metaheuristics. Although the latter produce better results in terms of minimizing the number of colors, the former are widely employed due to their simplicity. These heuristic methods are centralized since they operate by using complete knowledge of the graph. However, in real-world environmets where each component only interacts with a limited number of other components, the only option is to apply decentralized methods. This paper explores a novel and simple algorithm for decentralized graph coloring that uses a fixed number of colors and iteratively reduces the edge conflicts in the graph. We experimentally demonstrate that, for most of the tested instances, the new algorithm outperforms a recent and very competitive algorithm for decentralized graph coloring in terms of coloring quality. In our experiments, the fixed number of colors used by the new algorithm is controlled in a centralized manner.     

On characterizations of unimodality of discrete distributions

Annals of the Institute of Statistical Mathematics -     

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 discrete and continuous quotient Riesz spaces

Order properties of quotient Riesz spaces E/N(f) by null ideals N(f) are investigated. We show relationships between properties of a Riesz space E and its order dual E ^~ and properties of quotients E/N(f) where f runs over some subspaces of E ^~. A characterization of metrizable locally convex topological Riesz spaces whose all quotients (by proper closed ideals) are discrete is also given.     

On continuous wavelet transforms of distributions

Wavelet analysis is a universal and promising tool with very rich mathematical content and great potential for applications in various scientific fields, in particular, in signal (image) processing and the theory of differential equations. On the other hand distributions are widely used in these fields. And to apply wavelet analysis in these areas it is important to define and investigate wavelet transforms of distributions. In this paper we introduce continuous wavelet transforms of distributions and study convergence properties of these transforms.     

On the equivalence between some discrete and continuous optimization problems

The simplex algorithm for linear programming is based on the well-known equivalence between the problem of maximizing a linear functionf on a polyhedronP and the problem of maximizingf over the setV _P of all vertices ofP. The equivalence between these two problems is also exploited by some methods for maximizing a convex or quasi-convex function on a polyhedron.In this paper we determine some very general conditions under which the problem of maximizingf overP is equivalent, in some sense, to the problem of maximizingf overV _P . In particular, we show that these two problems are equivalent whenf is convex or quasi-convex on all the line segments contained inP and parallel to some edge ofP.In the case whereP is a box our results extend a well-known result of Rosenberg for 0–1 problems. Furthermore, whenP is a box or a simplex, we determine some classes of functions that can be maximized in polynomial time overP.This paper has been partially written while the author was visiting the Rutgers Center for Operations Research (RUTCOR). The support of the Air Force grants AFORS-89-0512 and AFORS-90-0008 is gratefully acknowledged.     

An asymptotic result on principal points for univariate distributions

We give an asymptotic result for principal points of univariate distributions, as defined in Flury (1990). Principal points are a generalization of the mean and provide a natural way to approximate a continuous distribution. We show that for a given density $si:f$esi:it is asymptotically optimal to take the quantiles of the density proportional to$si:f$esi:i^1/3 as principal points.