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Axioms
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New Trends in Discrete Probability and Statistics
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New Trends in Discrete Probability and Statistics

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: closed (28 February 2025) | Viewed by 4158

Special Issue Editor

Dr. Jean Renaud Pycke

E-Mail Website
Guest Editor
Département de Mathématiques, Université d'Evry Val d'Essonne, 91000 Evry, France
Interests: Bahadur efficiency; orthogonal polynomials and functions of hypergeometric types (Jacobi, Laguerre, Hermite, Askey scheme, etc.); Fourier series in special orthogonal functions; asymptotic properties of parametric tests; applications of statistics to biology and medical sciences; directional data; spatial statistics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues, 

The status of discrete statistics within the field of mathematical statistics is multiple and somewhat paradoxical. Most of the time, the collection of real data yields discrete empirical distributions. However, continuous models have received more attention, and consequently, more development than discrete ones. A first reason was already pointed out by Kendall and Stewart in their classic, The Advanced Theory of Statistics: ”In the ordinary data of experience our distributions are invariably discontinuous, because our measurements can only attain a certain degree of accuracy”; however, ”continuous distributions are generally amenable to more elegant mathematical treatment than are discrete distributions.” In their more recent encyclopedic book, Univariate Discrete Distributions, Johnson et al. confirmed this state of affairs: ”Goodness-of-fit tests for discrete distributions have not been researched as extensively as continuous distributions”. However, new prospects for discrete statistics are on the agenda, in the same way as how A. Terras describes for a Fourier analysis on groups and applications the renewal of discrete geometry: ”In this age of computer, it is very natural to replace the continuous with the finite. (...) Here in finite land, we will have no worries about convergence of integrals or interchange of summation and integration. Such worries often obscured the continuous theory in a myriad of analytical details.” Our aim is to illustrate these various aspects by means of different tools. 

Dr. Jean Renaud Pycke
Guest Editor

Manuscript Submission Information

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Axioms is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • parametric discrete distributions
  • distributions associated with discrete orthogonal polynomials
  • empirical distributions
  • empirical processes

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Published Papers (3 papers)

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Research

33 pages, 1390 KiB  
Article
Probabilistic Perturbation Bounds for Invariant, Deflating and Singular Subspaces
by Petko H. Petkov
Axioms 2024, 13(9), 597; https://doi.org/10.3390/axioms13090597 - 2 Sep 2024
Cited by 1 | Viewed by 884
Abstract
In this paper, we derive new probabilistic bounds on the sensitivity of invariant subspaces, deflation subspaces and singular subspaces of matrices. The analysis exploits a unified method for deriving asymptotic perturbation bounds of the subspaces under interest and utilizes probabilistic approximations of the [...] Read more.
In this paper, we derive new probabilistic bounds on the sensitivity of invariant subspaces, deflation subspaces and singular subspaces of matrices. The analysis exploits a unified method for deriving asymptotic perturbation bounds of the subspaces under interest and utilizes probabilistic approximations of the entries of random perturbation matrices implementing the Markoff inequality. As a result of the analysis, we determine with a prescribed probability asymptotic perturbation bounds on the angles between the corresponding perturbed and unperturbed subspaces. It is shown that the probabilistic asymptotic bounds proposed are significantly less conservative than the corresponding deterministic perturbation bounds. The results obtained are illustrated by examples comparing the known deterministic perturbation bounds with the new probabilistic bounds. Full article
(This article belongs to the Special Issue New Trends in Discrete Probability and Statistics)
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23 pages, 361 KiB  
Article
A Discrete Cramér–Von Mises Statistic Related to Hahn Polynomials with Application to Goodness-of-Fit Testing for Hypergeometric Distributions
by Jean-Renaud Pycke
Axioms 2024, 13(6), 369; https://doi.org/10.3390/axioms13060369 - 30 May 2024
Viewed by 675
Abstract
We give the Karhunen–Loève expansion of the covariance function of a family of discrete weighted Brownian bridges, appearing as discrete analogues of continuous Gaussian processes related to Cramér –von Mises and Anderson–Darling statistics. This analogy enables us to introduce a discrete Cramér–von Mises [...] Read more.
We give the Karhunen–Loève expansion of the covariance function of a family of discrete weighted Brownian bridges, appearing as discrete analogues of continuous Gaussian processes related to Cramér –von Mises and Anderson–Darling statistics. This analogy enables us to introduce a discrete Cramér–von Mises statistic and show that this statistic satisfies a property of local asymptotic Bahadur optimality for a statistical test involving the classical hypergeometric distributions. Our statistic and the goodness-of-fit problem we deal with are based on basic properties of Hahn polynomials and are, therefore, subject to some extension to all families of classical orthogonal polynomials, as well as their q-analogues. Due probably to computational difficulties, the family of discrete Cramér–von Mises statistics has received less attention than its continuous counterpart—the aim of this article is to bridge part of this gap. Full article
(This article belongs to the Special Issue New Trends in Discrete Probability and Statistics)
17 pages, 1463 KiB  
Article
Proportional Odds Hazard Model for Discrete Time-to-Event Data
by Maria Gabriella Figueiredo Vieira, Marcílio Ramos Pereira Cardial, Raul Matsushita and Eduardo Yoshio Nakano
Axioms 2023, 12(12), 1102; https://doi.org/10.3390/axioms12121102 - 6 Dec 2023
Cited by 3 | Viewed by 1834
Abstract
In this article, we present the development of the proportional odds hazard model for discrete time-to-event data. In this work, inferences about the model’s parameters were formulated considering the presence of right censoring and the discrete Weibull and log-logistic distributions. Simulation studies were [...] Read more.
In this article, we present the development of the proportional odds hazard model for discrete time-to-event data. In this work, inferences about the model’s parameters were formulated considering the presence of right censoring and the discrete Weibull and log-logistic distributions. Simulation studies were carried out to check the asymptotic properties of the estimators. In addition, procedures for checking the proportional odds assumption were proposed, and the proposed model is illustrated using a dataset on the survival time of patients with low back pain. Full article
(This article belongs to the Special Issue New Trends in Discrete Probability and Statistics)
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