Recent developments on the construction of bivariate distributions with fixed marginals

Given two discrete random variables X and Y , with probability distributions p = (p 1 , . . . , p n ) and q = (q 1 , . . . , q m ), respectively, denote by C(p, q) the set of all couplings of p and q, that is, the set of all bivariate probability distributions that have p and q as marginals. In this paper, we study the problem of finding the joint probability distribution in C(p, q) of minimum entropy (equivalently, the joint probability distribution that maximizes the mutual information between X and Y ), and we discuss several situations where the need for this kind of optimization naturally arises. Since the optimization problem is known to be NP-hard, we give an efficient algorithm to find a joint probability distribution in C(p, q) with entropy exceeding the minimum possible by at most 1, thus providing an approximation algorithm with additive approximation factor of 1.Leveraging on this algorithm, we extend our result to the problem of finding a minimum-entropy joint distribution of arbitrary k ≥ 2 discrete random variables X 1 , . . . , X k , consistent with the known k marginal distributions of X 1 , . . . , X k . In this case, our approximation algorithm has an additive approximation factor of log k.We also discuss some related applications of our findings.

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“…We have established the first equation of (14). Moreover, the second equation of (14) also holds because by the equality in (16)…”

Section: A the Analysis Of Correctness Of Algorithm 1: Technical Lemmasmentioning

confidence: 99%

How to find a joint probability distribution of minimum entropy (almost) given the marginals

Cicalese

Gargano

Vaccaro

2017

2017 IEEE International Symposium on Information Theory (ISIT)

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“…A goal of this study was to find a bivariate distribution jointly representing the number of pods and stems with white mould, while accounting for the non‐zero correlation between the two sets of counts; a plant having pods with white mould was also likely to have stems with white mould. Many approaches to construct a bivariate distribution have been described (Lin et al ., ). A simple starting point would have been the Farlie‐Gumbel‐Morgenstern (FGM) family of copulas (Nelsen, ), but FGM bivariate distributions suffer from being restricted to a maximum allowable correlation of one‐third to be a legitimate bivariate distribution.…”

Section: Discussionmentioning

confidence: 97%

“…Lee () highlighted the Sarmanov family of distributions (Sarmanov, ), which have seen a resurgence in interest (Hofer & Leitner, ; Pelican & Vernic, ; Lin et al ., ). For the white mould data, a Sarmanov bivariate distribution with NB marginals was derived, and the fitted parameter ω satisfied the required boundary constraints.…”

Section: Discussionmentioning

confidence: 97%

Probability distributions for marketable pods and white mould on snap bean

Shah

Dillard

Pethybridge

2017

Annals of Applied Biology

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White mould, caused by Sclerotinia sclerotiorum, is one of the most recalcitrant diseases of snap bean. Probability distributions suitable for describing the total number of marketable pods (hereafter simply referred to as pods) per plant, as well as for the number of pods and stems with white mould per plant, have not been identified. The total number of pods and the number of pods and stems with white mould were measured on a per plant basis in plots of processing snap bean (var. Hystyle) in New York. The total number of pods per plant ranged from 0 to 29, and was best described by the Pólya-Aeppli distribution. The number of pods and the number of stems with white mould per plant were well-described by the negative binomial (NB) distribution. A Sarmanov bivariate distribution with NB marginals was derived and fitted to the joint data on the number of pods and stems with white mould per plant, accounting for correlation between pods and stems with white mould on the same plant. The bivariate distribution was used to formulate an empirical equation for the incidence of plants with white mould as a function of the average number of pods and stems with white mould per plant. The results represented a more complete understanding of the distributional properties of white mould in snap bean.

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“…The cdf of T k:n s , which is the lifetime of an (n − k + 1)-outof-n ARSL system involves the joint distribution of bivariate order statistics (X r:n , Y s:n) , 1 ≤ r < s ≤ n. For some recent results on bivariate order statistics and their connection with bivariate binomial distributions see [6,7,19,23]. …”

Section: The Reliability Of a Coherent System With An Active Redundanmentioning

confidence: 99%

Reliability and mean residual life functions of coherent systems in an active redundancy

Kavlak

2017

Naval Research Logistics

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Abstract:In this article, the reliability and the mean residual life (MRL) functions of a system with active redundancies at the component and system levels are investigated. In active redundancy at the component level, the original and redundant components are working together and lifetime of the system is determined by the maximum of lifetime of the original components and their spares. In the active redundancy at the system level, the system has a spare, and the original and redundant systems work together. The lifetime of such a system is then the maximum of lifetimes of the system and its spare. The lifetimes of the original component and the spare are assumed to be dependent random variables

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Recent developments on the construction of bivariate distributions with fixed marginals

Cited by 17 publications

References 51 publications

How to find a joint probability distribution of minimum entropy (almost) given the marginals

How to find a joint probability distribution of minimum entropy (almost) given the marginals

Probability distributions for marketable pods and white mould on snap bean

Reliability and mean residual life functions of coherent systems in an active redundancy

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