Mathematical Tools and Techniques Applicable to Probability Theory and Statistics II

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

Deadline for manuscript submissions: 30 January 2025 | Viewed by 8251

Special Issue Editor


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Guest Editor
Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada
Interests: real and complex analysis; fractional calculus and its applications; integral equations and transforms; higher transcendental functions and their applications; q-series and q-polynomials; analytic number theory; analytic and geometric Inequalities; probability and statistics; inventory modelling and optimization
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Special Issue Information

Dear Colleagues,

This issue is a continuation of the previous successful Special Issue entitled “Mathematical Tools and Techniques Applicable to Probability Theory and Statistics”. Investigations involving the theory and applications of mathematical analytic tools and techniques are remarkably widespread in many diverse areas of the mathematical, physical, biological, chemical, engineering, and statistical sciences. In this Special Issue, we invite and welcome review, expository, and original research articles dealing with (but not limited to) the recent advances in the subject of (among other related areas) probability theory and statistics.

We are looking forward to your contribution to this Special Issue.

Prof. Dr. Hari Mohan Srivastava
Guest Editor

Manuscript Submission Information

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Keywords

  • mathematical (or higher transcendental) functions and their applications in probability theory and statistics
  • probabilistic derivations and applications of generating functions
  • the notion of statistical convergence and related developments
  • stochastic and martingale sequences and associated approximation theorems
  • statistical inference, statistical mechanics and related areas
  • summability theory and statistical applications

Published Papers (6 papers)

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Research

18 pages, 1324 KiB  
Article
Assessment of the Water Distribution Networks in the Kingdom of Saudi Arabia: A Mathematical Model
by Aiman Albarakati, Asifa Tassaddiq and Rekha Srivastava
Axioms 2023, 12(11), 1055; https://doi.org/10.3390/axioms12111055 - 16 Nov 2023
Viewed by 1050
Abstract
Graph theory is a branch of mathematics that is crucial to modelling applicable systems and networks using matrix representations. In this article, a novel graph-theoretic model was used to assess an urban water distribution system (WDS) in Saudi Arabia. This graph model is [...] Read more.
Graph theory is a branch of mathematics that is crucial to modelling applicable systems and networks using matrix representations. In this article, a novel graph-theoretic model was used to assess an urban water distribution system (WDS) in Saudi Arabia. This graph model is based on representing its elements through nodes and links using a weighted adjacency matrix. The nodes represent the points where there can be a water input or output (sources, treatment plants, tanks, reservoirs, consumers, connections), and links represent the edges of the graph that carry water from one node to another (pipes, pumps, valves). Four WDS benchmarks, pumps, tanks, reservoirs, and external sources were used to validate the framework at first. This validation showed that the worst-case scenarios for vulnerability were provided by the fault sequence iterating the calculation of the centrality measurements. The vulnerability framework’s application to the Saudi Arabian WDS enabled the identification of the system’s most vulnerable junctions and zones. As anticipated, the regions with the fewest reservoirs were most at risk from unmet demand, indicating that this system is vulnerable to the removal of junctions and pipes that are intricately associated with their neighbours. Different centrality metrics were computed, from which the betweenness centrality offered the worst vulnerability prediction measures. The aspects and zones of the WDS that can more significantly impact the water supply in the event of a failure were identified by the vulnerability framework utilising attack tactics. Full article
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19 pages, 425 KiB  
Article
A Certain Class of Equi-Statistical Convergence in the Sense of the Deferred Power-Series Method
by Hari Mohan Srivastava, Bidu Bhusan Jena and Susanta Kumar Paikray
Axioms 2023, 12(10), 964; https://doi.org/10.3390/axioms12100964 - 13 Oct 2023
Viewed by 970
Abstract
In this paper, we expose the ideas of point-wise statistical convergence, equi-statistical convergence and uniform statistical convergence in the sense of the deferred power-series method. We then propose a relation connecting them, which is followed by several illustrative examples. Moreover, as an application [...] Read more.
In this paper, we expose the ideas of point-wise statistical convergence, equi-statistical convergence and uniform statistical convergence in the sense of the deferred power-series method. We then propose a relation connecting them, which is followed by several illustrative examples. Moreover, as an application viewpoint, we establish an approximation theorem based upon our proposed method for equi-statistical convergence of sequences of positive linear operators. Finally, we estimate the equi-statistical rates of convergence for the effectiveness of the results presented in our study. Full article
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22 pages, 9590 KiB  
Article
Bivariate Discrete Odd Generalized Exponential Generator of Distributions for Count Data: Copula Technique, Mathematical Theory, and Applications
by Laila A. Al-Essa, Mohamed S. Eliwa, Hend S. Shahen, Amal A. Khalil, Hana N. Alqifari and Mahmoud El-Morshedy
Axioms 2023, 12(6), 534; https://doi.org/10.3390/axioms12060534 - 29 May 2023
Viewed by 896
Abstract
In this article, a new family of bivariate discrete distributions is proposed based on the copula concept, in the so-called bivariate discrete odd generalized exponential-G family. Some distributional properties, including the joint probability mass function, joint survival function, joint failure rate function, median [...] Read more.
In this article, a new family of bivariate discrete distributions is proposed based on the copula concept, in the so-called bivariate discrete odd generalized exponential-G family. Some distributional properties, including the joint probability mass function, joint survival function, joint failure rate function, median correlation coefficient, and conditional expectation, are derived. After proposing the general class, one special model of the new bivariate family is discussed in detail. The maximum likelihood approach is utilized to estimate the family parameters. A detailed simulation study is carried out to examine the bias and mean square error of maximum likelihood estimators. Finally, the importance of the new bivariate family is explained by means of two distinctive real data sets in various fields. Full article
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12 pages, 1807 KiB  
Article
Consistency of the Estimator for the Common Mean in Fixed-Effect Meta-Analyses
by Nanami Taketomi and Takeshi Emura
Axioms 2023, 12(5), 503; https://doi.org/10.3390/axioms12050503 - 21 May 2023
Cited by 1 | Viewed by 1350
Abstract
Fixed-effect meta-analyses aim to estimate the common mean parameter by the best linear unbiased estimator. Besides unbiasedness, consistency is one of the most fundamental requirements for the common mean estimator to be valid. However, conditions for the consistency of the common mean estimator [...] Read more.
Fixed-effect meta-analyses aim to estimate the common mean parameter by the best linear unbiased estimator. Besides unbiasedness, consistency is one of the most fundamental requirements for the common mean estimator to be valid. However, conditions for the consistency of the common mean estimator have not been discussed in the literature. This article fills this gap by clarifying conditions for making the common mean estimator consistent in fixed-effect meta-analyses. In this article, five theorems are devised, which state regularity conditions for the common mean estimator to be consistent. These theorems are novel applications of the classical large sample theory to meta-analyses. Numerical illustrations are also given to help understand the needs of the regularity conditions. Three real datasets illustrate the practical consequences of the devised theorems. This article concludes that the inconsistency of the common mean estimator occurs under some conditions in real meta-analyses. Full article
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26 pages, 5485 KiB  
Article
Bayesian Inference and Data Analysis of the Unit–Power Burr X Distribution
by Aisha Fayomi, Amal S. Hassan, Hanan Baaqeel and Ehab M. Almetwally
Axioms 2023, 12(3), 297; https://doi.org/10.3390/axioms12030297 - 14 Mar 2023
Cited by 13 | Viewed by 1304
Abstract
The unit–power Burr X distribution (UPBXD), a bounded version of the power Burr X distribution, is presented. The UPBXD is produced through the inverse exponential transformation of the power Burr X distribution, which is also beneficial for modelling data on the unit interval. [...] Read more.
The unit–power Burr X distribution (UPBXD), a bounded version of the power Burr X distribution, is presented. The UPBXD is produced through the inverse exponential transformation of the power Burr X distribution, which is also beneficial for modelling data on the unit interval. Comprehensive analysis of its key characteristics is performed, including shape analysis of the primary functions, analytical expression for moments, quantile function, incomplete moments, stochastic ordering, and stress–strength reliability. Rényi, Havrda and Charvat, and d-generalized entropies, which are measures of uncertainty, are also obtained. The model’s parameters are estimated using a Bayesian estimation approach via symmetric and asymmetric loss functions. The Bayesian credible intervals are constructed based on the marginal posterior distribution. Monte Carlo simulation research is intended to test the accuracy of various estimators based on certain measures, in accordance with the complex forms of Bayesian estimators. Finally, we show that the new distribution is more appropriate than certain other competing models, according to their application for COVID-19 in Saudi Arabia and the United Kingdom. Full article
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24 pages, 1332 KiB  
Article
Different Estimation Methods for New Probability Distribution Approach Based on Environmental and Medical Data
by Eid A. A. Hassan, Mohammed Elgarhy, Eman A. Eldessouky, Osama H. Mahmoud Hassan, Essam A. Amin and Ehab M. Almetwally
Axioms 2023, 12(2), 220; https://doi.org/10.3390/axioms12020220 - 20 Feb 2023
Cited by 5 | Viewed by 1789
Abstract
In this article, we introduce a new extension of the power Lomax (PLo) model by combining the type II exponentiated half-logistic class of statistical models and the PLo model. The new suggested statistical model called type II exponentiated half-logistic-PLo (TIIEHL-PLo) model. However, the [...] Read more.
In this article, we introduce a new extension of the power Lomax (PLo) model by combining the type II exponentiated half-logistic class of statistical models and the PLo model. The new suggested statistical model called type II exponentiated half-logistic-PLo (TIIEHL-PLo) model. However, the new TIIEHL-PLo model is more flexible and applicable than the PLo model and some extensions of THE PLo model, especially those in environmental and medical fields. Some general statistical properties of the TIIEHL-PLo model are computed. Six different estimation approaches, namely maximum likelihood (ML), least-square (LS), weighted least-squares (WLS), maximum product spacing (MPS), Cramér–von Mises (CVM), and Anderson–Darling (AD) estimation approaches, are utilized to estimate the parameters of the TIIEHL-PLo model. The simulation experiment examines the accuracy of the model parameters by employing six different methodologies of estimation. In this study, we analyze three real datasets from the environmental and medical fields to highlight the relevance and adaptability of the proposed approach. The newly suggested model is exceptionally adaptable and outperforms several well-known statistical models. Full article
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