Negative dependence and stochastic orderings

Daly, Fraser

doi:10.1051/ps/2016002

Cited by 4 publications

(6 citation statements)

References 28 publications

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“…In (ii), the lower bound is standard and the upper bound is from Theorem 1.1 of Daly and Johnson [14]. (iii) follows from Theorem 2.2 and Corollary 2.8 of Daly [13].…”

Section: Ordering Results For Cmp Distributionsmentioning

confidence: 99%

“…However, in the case ν < 1 we cannot adapt the proof of Theorem 1.1 of Daly and Johnson [14] to give an analogue of Proposition 2.12 (ii). By suitably modifying the proof of Theorem 2.2 of Daly [13], we may note that Z ≤ cx X in this case, where Z ∼ Po(µ). There is, however, no concentration inequality corresponding to that given in Proposition 2.12 (iii).…”

Section: Ordering Results For Cmp Distributionsmentioning

confidence: 99%

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The Conway-Maxwell-Poisson distribution: Distributional theory and approximation

Daly¹,

Gaunt²

2016

ALEA

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The Conway-Maxwell-Poisson (CMP) distribution is a natural two-parameter generalisation of the Poisson distribution which has received some attention in the statistics literature in recent years by offering flexible generalisations of some well-known models. In this work, we begin by establishing some properties of both the CMP distribution and an analogous generalisation of the binomial distribution, which we refer to as the CMB distribution. We also consider some convergence results and approximations, including a bound on the total variation distance between a CMB distribution and the corresponding CMP limit. A two-parameter generalisation of the Poisson distribution was introduced by Conway and Maxwell [10] as the stationary number of occupants of a queuing system with state dependent service or arrival rates. This distribution has since become known as the Conway-Maxwell-Poisson (CMP) distribution. Beginning with the work of Boatwright, Borle and Kadane [7] and Shmueli et al. [31], the CMP distribution has received recent attention in the statistics literature on account of the flexibility it offers in statistical models. For example, the CMP distribution can model data which is either under-or over-dispersed relative to the Poisson distribution. This property is exploited by Sellers and Shmueli [29], who use the CMP distribution to generalise the Poisson and logistic regression models. Kadane et al. [24] considered the use of the CMP distribution in Bayesian analysis, and Wu, Holan and Wilkie [34] use the CMP distribution as part of a

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“…In (ii), the lower bound is standard and the upper bound is from Theorem 1.1 of Daly and Johnson [14]. (iii) follows from Theorem 2.2 and Corollary 2.8 of Daly [13].…”

Section: Ordering Results For Cmp Distributionsmentioning

confidence: 99%

Section: Ordering Results For Cmp Distributionsmentioning

confidence: 99%

The Conway-Maxwell-Poisson distribution: Distributional theory and approximation

Daly¹,

Gaunt²

2016

ALEA

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“…Property P3 provides a form of negative association. It is weaker than the usual form of negative association [which corresponds to the analogue of (3.2) with i and Λ \ {i} replaced by arbitrary disjoint subsets A, B ⊂ Λ], but stronger than other related notions, such as totally negative dependence (see [11] for this and other forms of negative dependence).…”

Section: A Representation Of Fields Satisfying (Fe) and (Fkg)mentioning

confidence: 94%

A quantitative Burton–Keane estimate under strong FKG condition

Duminil-Copin¹,

Ioffe²,

Velenik³

2016

Ann. Probab.

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We consider translationally-invariant percolation models on Z d satisfying the finite energy and the FKG properties. We provide explicit upper bounds on the probability of having two distinct clusters going from the endpoints of an edge to distance n (this corresponds to a finite size version of the celebrated Burton-Keane [Comm. Math. Phys. 121 (1989) 501-505] argument proving uniqueness of the infinite-cluster). The proof is based on the generalization of a reverse Poincaré inequality proved in Chatterjee and Sen (2013). As a consequence, we obtain upper bounds on the probability of the so-called four-arm event for planar random-cluster models with cluster-weight q ≥ 1.

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“…This is a stronger assumption than that P * is stochastically dominated by P (write P * ≤ st P ) -see [82] for a review of stochastic ordering results. As described in [34,35,75], such orderings naturally arise in many contexts where P is the mass function of a sum of negatively dependent random variables.…”

Section: Technical Definitions In the Discrete Casementioning

confidence: 99%

“…We would like to introduce concavity conditions mirroring Conditions 1 or 2, under which results of the form Theorem 5.4 and 5.5 can be proved for the derivative ∇ n of (25). In addition, it would be of interest to see whether a version of Theorem 5.5 holds under a stochastic ordering assumption P * ≤ st P , in the spirit of [34].…”

Section: [Concentration Inequalities]mentioning

confidence: 99%

Entropy and Thinning of Discrete Random Variables

Johnson

2017

Convexity and Concentration

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We describe five types of results concerning information and concentration of discrete random variables, and relationships between them, motivated by their counterparts in the continuous case. The results we consider are information theoretic approaches to Poisson approximation, the maximum entropy property of the Poisson distribution, discrete concentration (Poincaré and logarithmic Sobolev) inequalities, monotonicity of entropy and concavity of entropy in the Shepp-Olkin regime.

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Negative dependence and stochastic orderings

Cited by 4 publications

References 28 publications

The Conway-Maxwell-Poisson distribution: Distributional theory and approximation

The Conway-Maxwell-Poisson distribution: Distributional theory and approximation

A quantitative Burton–Keane estimate under strong FKG condition

Entropy and Thinning of Discrete Random Variables

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