Uniform order statistics property and ℓ<sub>∞</sub>-spherical densities

Shaked, Moshe; Spizzichino, Fabio; Suter, Florentina

doi:10.1017/s0269964808000181

Cited by 4 publications

(27 citation statements)

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“…This paper is in fact concentrated on the theme of Exchangeable Occupancy Models (EOM) and their relations with the Uniform Order Statistics Property (UOSP) of counting processes in discrete time. As one main purpose of ours, we show that some notions and results given in Shaked, Spizzichino and Suter [11,12] admit completely natural generalizations in the frame of EOM's.After appropriate preliminaries, we will consider some general properties of the EOM's. For our purposes, we then introduce the notion of M (a) -models, a relevant sub-class of EOM's that turns out to have an important role in our derivations.…”

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confidence: 64%

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confidence: 64%

“…Our work arises naturally as an attempt to generalize the definitions of UOSP presented in Shaked et al [11,12]. For sake of readability and to make the setting clear, first we briefly summarize main facts of interest therein contained.…”

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confidence: 99%

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Exchangeable Occupancy Models and Discrete Processes With the Generalized Uniform Order Statistics Property

Collet¹,

Leisen²,

Spizzichino³

et al. 2013

Prob. Eng. Inf. Sci.

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This work focuses on Exchangeable Occupancy Models (EOM) and their relations with the Uniform Order Statistics Property (UOSP) for point processes in discrete time. As our main purpose, we show how definitions and results presented in Shaked, Spizzichino and Suter [11] can be unified and generalized in the frame of occupancy models. We first show some general facts about EOM's. Then we introduce a class of EOM's, called M (a) -models, and a concept of generalized Uniform Order Statistics Property in discrete time. For processes with this property, we prove a general characterization result in terms of M (a) -models. Our interest is also focused on properties of closure w.r.t. some natural transformations of EOM's.The so called occupancy distributions give rise, as well known, to a class of multivariate models useful in the description of randomized phenomena. The name "occupancy" comes from the interpretation in terms of particles that are randomly distributed among several cells. In particular, three classical examples, related to as many well-known physical systems, belong to this class: Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac model, see Feller [5]. In the Maxwell-Boltzmann (MB) statistics the capacity of each cell is unlimited, and the particles are distinguishable. In the Bose-Einstein (BE) statistics, the capacity of each cell is unlimited but the particles are indistinguishable. In the Fermi Dirac (FD) statistics, the particles are indistinguishable, but cells can only hold a maximum of one particle. All these three statistics assume that the cells are distinguishable. These models are attractive for many reasons. First of all they have wide range of applications in Sciences, Engineering and also in Statistics, as pointed out by Charalambides [2, Chapters 4 and 5] and Gadrich and Ravid [6]. Moreover Mahmoud [10] provided an interpretation of occupancy distributions in terms of Pólya Urns, which are very flexible and applicable to problems arising in various areas; e.g. Clinical Trials (see Crimaldi and Leisen [4] for some references), Economics (see Aruka [1]) and Computer Science (see Shah, Kothari, Jayadeva andChandra [13]). From a probabilistic and combinatoric point of view, the three fundamental models (MB, BE and FD) have many interesting properties. Indeed they are, in particular, exchangeable and this is a basic remark for our aims. This paper is in fact concentrated on the theme of Exchangeable Occupancy Models (EOM) and their relations with the Uniform Order Statistics Property (UOSP) of counting processes in discrete time. As one main purpose of ours, we show that some notions and results given in Shaked, Spizzichino and Suter [11,12] admit completely natural generalizations in the frame of EOM's.After appropriate preliminaries, we will consider some general properties of the EOM's. For our purposes, we then introduce the notion of M (a) -models, a relevant sub-class of EOM's that turns out to have an important role in our derivations. In the final part of the paper, we will introduce ...

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Exchangeable Occupancy Models and Discrete Processes With the Generalized Uniform Order Statistics Property

Collet¹,

Leisen²,

Spizzichino³

et al. 2013

Prob. Eng. Inf. Sci.

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“…Now, by Theorem 2.1, (3.2) follows. If w is differentiable and decreasing, Shaked et al [33] proved that there exists a random variable T nþ1 such that T 1 Á Á Á T n T nþ1 have an ' 1 4 -spherical density of the form…”

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confidence: 99%

“…For a modified mixed geometric process, it is seen from Theorem 4.7 in Shaked et al[33] that the first n epoch times T 1 , . .…”

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Nonnegativity of Covariances Between Functions of Ordered Random Variables

Yao

2007

Prob. Eng. Inf. Sci.

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In this article we investigate conditions by a unified method under which the covariances of functions of two adjacent ordered random variables are nonnegative. The main structural results are applied to several kinds of ordered random variable, such as delayed record values, continuous and discrete ' 1 4 -spherical order statistics, epoch times of mixed Poisson processes, generalized order statistics, discrete weak record values, and epoch times of modified geometric processes. These applications extend the main results for ordinary order statistics in Qi [28] and for usual record values in Nagaraja and Nevzorov [25].

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Merging Exchangeable Occupancy Distributions: The Family 𝓜 ( a ) $\mathcal {M}^{(a)}$ and its Connection with the Maximum Entropy Principle

Collet

Leisen

Spizzichino

2015

Methodol Comput Appl Probab

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In this paper a new transformation of occupancy distributions, called merging, is introduced. In particular, it will be studied the effect of merging on a class of occupancy distributions that was recently introduced in Collet et al. (Probab Eng Inf Sci. 27:533-552 2013). These results have an interesting interpretation in the so-called entropy maximization inference. The last part of the paper is devoted to highlight the impact of our findings in this research area.

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Uniform order statistics property and ℓ_∞-spherical densities

Cited by 4 publications

References 1 publication

Exchangeable Occupancy Models and Discrete Processes With the Generalized Uniform Order Statistics Property

Exchangeable Occupancy Models and Discrete Processes With the Generalized Uniform Order Statistics Property

Nonnegativity of Covariances Between Functions of Ordered Random Variables

Merging Exchangeable Occupancy Distributions: The Family 𝓜 ( a ) $\mathcal {M}^{(a)}$ and its Connection with the Maximum Entropy Principle

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Uniform order statistics property and ℓ∞-spherical densities

Cited by 4 publications

References 1 publication

Exchangeable Occupancy Models and Discrete Processes With the Generalized Uniform Order Statistics Property

Exchangeable Occupancy Models and Discrete Processes With the Generalized Uniform Order Statistics Property

Nonnegativity of Covariances Between Functions of Ordered Random Variables

Merging Exchangeable Occupancy Distributions: The Family 𝓜 ( a ) $\mathcal {M}^{(a)}$ and its Connection with the Maximum Entropy Principle

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Uniform order statistics property and ℓ_∞-spherical densities