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Applied and Computational Statistics • Sorana D Applied and Computational Statistics Computational and Applied • Sorana D. Bolboacă D. Sorana • Applied and Computational Statistics Edited by Sorana D. Bolboacă Printed Edition of the Special Issue Published in Mathematics www.mdpi.com/journal/mathematics Applied and Computational Statistics Applied and Computational Statistics Special Issue Editor Sorana D. Bolboac˘a MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade Special Issue Editor Sorana D. Bolboaca˘ Iuliu Hat¸ieganu University of Medicine and Pharmacy Romania Editorial Office MDPI St. Alban-Anlage 66 4052 Basel, Switzerland This is a reprint of articles from the Special Issue published online in the open access journal Mathematics (ISSN 2227-7390) from 2018 to 2019 (available at: https://www.mdpi.com/journal/ mathematics/special issues/applied computational statistics). For citation purposes, cite each article independently as indicated on the article page online and as indicated below: LastName, A.A.; LastName, B.B.; LastName, C.C. Article Title. Journal Name Year, Article Number, Page Range. ISBN 978-3-03928-176-3 (Pbk) ISBN 978-3-03928-177-0 (PDF) c 2020 by the authors. Articles in this book are Open Access and distributed under the Creative Commons Attribution (CC BY) license, which allows users to download, copy and build upon published articles, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. The book as a whole is distributed by MDPI under the terms and conditions of the Creative Commons license CC BY-NC-ND. Contents About the Special Issue Editor ...................................... vii Preface to ”Applied and Computational Statistics” .......................... ix Ibrahim Elbatal, Farrukh Jamal, Christophe Chesneau, Mohammed Elgarhy and Sharifah Alrajhi The Modified Beta Gompertz Distribution: Theory and Applications Reprinted from: Mathematics 2019, 7, 3, doi:10.3390/math7010003 .................. 1 Lorentz J¨antschi and Sorana D. Bolboac˘a Computation of Probability Associated with Anderson–Darling Statistic Reprinted from: Mathematics 2018, 6, 88, doi:10.3390/math6060088 .................. 18 Miltiadis S. Chalikias Optimal Repeated Measurements for Two Treatment Designs with Dependent Observations: The Case of Compound Symmetry Reprinted from: Mathematics 2019, 7, 378, doi:10.3390/math7040378 ................. 35 Lili Tan, Yunzhan Gong and Yawen Wang A Model for Predicting Statement Mutation Scores Reprinted from: Mathematics 2019, 7, 778, doi:10.3390/math7090778 ................. 41 Dan-Marian Joit¸a and Lorentz J¨antschi Extending the Characteristic Polynomial for Characterization of C20 Fullerene Congeners Reprinted from: Mathematics 2017, 5, 84, doi:10.3390/math5040084 .................. 80 v About the Special Issue Editor Sorana D. Bolboac˘a is a professor of medical informatics and biostatistics at the ”Iuliu Hat, ieganu” University of Medicine and Pharmacy Cluj-Napoca, Romania. She earned her Ph.D. in Medicine (2006) from the Iuliu Hat¸ieganu University of Medicine and Pharmacy (title of the thesis Evidence-Based Medicine: Logistics and Implementation) and a Ph.D. in Horticulture (2010) from the University of Agriculture Sciences and Veterinary Medicine Cluj-Napoca (title of the thesis Statistical Models for Analysis of Genetic Variability). Her research interests are multidisciplinary, and include applied and computational statistics, molecular modeling, genetic analysis, statistical modeling in medicine, integrated health informatics systems and the application of new technologies in medicine, medical diagnostics research, medical imaging analysis, assisted decision systems, research ethics, social media and health information, and evidence-based medicine. She is the author of more than 200 papers and 19 monographs in medicine, computational chemistry, computer science, mathematics, environmental sciences, biomedical engineering, nanoscience nanotechnology, and medical informatics. vii Preface to ”Applied and Computational Statistics” The research on statistical populations, samples, or models have applications in all research areas and are conducted to gain knowledge for real-world problems. Increased calculation power opens the path to computational statistics, algorithm translation, and the implementation of statistical methods and computer simulations. These areas are developing rapidly, providing solutions to multidisciplinary, interdisciplinary, and transdisciplinary topics. An excellent theoretical statistics method is worthless in the absence of real-data applicability. Furthermore, any excellent theoretical statistics method will find its ending sooner or later without proper implementation. Statistical methods find their application is understanding phenomenon from all fields, including medicine, biology, biochemistry, agriculture, horticulture, engineering, and more. The Special Issue of Mathematics entitled “Applied and Computational Statistics” provides new methods and their applicability to the prediction of the mutation score, repeated measurements bi-treatment cross-over design, modified beta Gompertz distribution, extending the characteristic polynomial, and computation of the probability associated with Anderson–Darling statistics. In addition to giving a detailed presentation of the implemented method that assures the reproducibility, the collection of articles also includes specific applicability examples. Thanks to all contributors for their involvement in finding statistical solutions to real-life problems; keep staying on the path of knowledge gain. Dear reviewers, thank you very much for the time spent in reviewing the articles and for the constructive comments that have certainly contributed to the quality of the papers. I warmly invite readers to enjoy reading this collection of articles, and I hope that new ideas will come to life for the benefit of science by reading these manuscripts. Sorana D. Bolboac˘a Special Issue Editor ix mathematics Article The Modified Beta Gompertz Distribution: Theory and Applications Ibrahim Elbatal 1, Farrukh Jamal 2, Christophe Chesneau 3,*, Mohammed Elgarhy 4 and Sharifah Alrajhi 5 1 Institute of Statistical Studies and Research (ISSR), Department of Mathematical Statistics, Cairo University, Giza 12613, Egypt; [email protected] 2 Department of Statistics, Govt. S.A Postgraduate College Dera Nawab Sahib, Bahawalpur, Punjab 63360, Pakistan; [email protected] 3 Department of Mathematics, LMNO, University of Caen, 14032 Caen, France 4 Department of Statistics, University of Jeddah, Jeddah 21589, Saudi Arabia; [email protected] 5 Department of Statistics, Faculty of Sciences, King Abdulaziz University, Jeddah 21589, Saudi Arabia; [email protected] * Correspondence: [email protected]; Tel.: +33-02-3156-7424 Received: 7 November 2018; Accepted: 17 December 2018; Published: 20 December 2018 Abstract: In this paper, we introduce a new continuous probability distribution with five parameters called the modified beta Gompertz distribution. It is derived from the modified beta generator proposed by Nadarajah, Teimouri and Shih (2014) and the Gompertz distribution. By investigating its mathematical and practical aspects, we prove that it is quite flexible and can be used effectively in modeling a wide variety of real phenomena. Among others, we provide useful expansions of crucial functions, quantile function, moments, incomplete moments, moment generating function, entropies and order statistics. We explore the estimation of the model parameters by the obtained maximum likelihood method. We also present a simulation study testing the validity of maximum likelihood estimators. Finally, we illustrate the flexibility of the distribution by the consideration of two real datasets. Keywords: modified beta generator; gompertz distribution; maximum likelihood estimation MSC: 60E05; 62E15; 62F10 1. Introduction The Gompertz distribution is a continuous probability distribution introduced by Gompertz [1]. The literature about the use of the Gompertz distribution in applied areas is enormous. A nice review can be found in [2], and the references therein. From a mathematical point of view, the cumulative probability density function (cdf) of the Gompertz distribution with parameters λ > 0 and α > 0is given by − λ ( αx− ) G(x)=1 − e α e 1 , x > 0. The related probability density function (pdf) is given by α − λ ( αx− ) g(x)=λe xe α e 1 , x > 0. It can be viewed as a generalization of the exponential distribution (obtained with α → 0) and thus an alternative to the gamma or Weibull distribution. A feature of the Gompertz distribution is that g(x) is unimodal and has positive skewness, whereas the related hazard rate function (hrf) given by h(x)=g(x)/(1 − G(x)) is increasing. To increase the flexibility of the Gompertz distribution, further Mathematics 2019, 7, 3; doi:10.3390/math70100031 www.mdpi.com/journal/mathematics Mathematics 2019, 7,3 extensions have been proposed. A natural one is the generalized Gompertz distribution introduced by El-Gohary et al. [3]. By introducing an exponent parameter a > 0, the related cdf is given by − λ ( αx− ) a F(x)= 1 − e α e 1 , x > 0. The related applications show that a plays an important role in term of model flexibility. This idea was then extended by Jafari et al. [4] who used the so-called beta generator introduced by Eugene et al. [5]. The related cdf is given by λ α − − α (e x−1) 1 1 e − − F(x)= ta 1(1 − t)b 1dt B(a, b) 0 = ( ) > I − λ ( αx− ) a, b
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