The Unit Omega Distribution as an Alternative to the Beta Distribution for the Modeling to the Infrared Thermography Temperature Data

Authors

  • Alia A. Alkhathami Department of Basic Science, College of Science and Theoretical Studies, Saudi Electronic University, Riyadh 11673, Kingdom of Saudi Arabia
  • Mahfooz Alam Department of Mathematics and Statistics, Faculty of Science and Technology, Vishwakarma University, Pune-411 048, India
  • Aafaq A. Rather Symbiosis Statistical Institute, Symbiosis International (Deemed University), Pune, India
  • Mehdi Hosseinzadeh Department of AI, School of Computer Science and Engineering, Galgotias University, Greater Noida, India, International Center for Materials Sciences and Technology, Western Caspian University, Baku, Azerbaijan
  • Showkat Ahmad Lone Department of Basic Science, College of Science and Theoretical Studies, Saudi Electronic University, Riyadh 11673, Kingdom of Saudi Arabia

DOI:

https://doi.org/10.13052/jrss0974-8024.19210

Keywords:

Order Statistics, Record Values, L-Moments, Moments, Recurrence relations, Absolute Stirling number, Infrared Thermography Temperature

Abstract

Proportion data is significant in many disciplines, including as economics, finance, reliability engineering, medicine, biology, and chemistry, since it serves as the basis for identifying trends, expediting procedures, and making well-informed conclusions that result in advances and innovations. For the representation and analysis of proportional data, the beta distribution is a standard model. In this paper, we explore the unit omega distribution [15] and their ordered properties such as the exact expressions, as well as recurrence relations, for the single moments of the order statistics (OSs) and record values. Additionally, various L-moment characteristics based on OS moments and record values were analyzed. The applicability of the unit omega distribution over beta distribution was demonstrated through its successful fit to the original IRT temperature dataset. For comparative assessment, the performance of the unit omega distribution was evaluated against the commonly used beta distribution using the Kolmogorov–Smirnov (KS) goodness-of-fit test. The results indicate that the unit omega distribution yields a smaller KS test statistic and a larger associated p-value compared to the Beta distribution, suggesting a superior fit to the observed data. These findings highlight the flexibility and effectiveness of the unit omega distribution in modelling bounded data. Furthermore, the study provides a strong foundation for future research, particularly in extending the analysis to generalized order statistics and progressive censoring schemes associated with the unit omega distribution.

Downloads

Download data is not yet available.

Author Biographies

Alia A. Alkhathami, Department of Basic Science, College of Science and Theoretical Studies, Saudi Electronic University, Riyadh 11673, Kingdom of Saudi Arabia

Alia A. Alkhathami is an Assistant Professor of Mathematics and Statistics at Saudi Electronic University (SEU), Kingdom of Saudi Arabia. She earned her M.Sc. in Probability and Statistics, from Carleton University, Canada, in 2015 and her Ph.D. in Probability and Statistics, from Carleton University, Canada, in 2021. Her research interests include Statistics, missing data analysis, longitudinal data analysis and applied statistical modeling, and she has published extensively in reputed international journals. She actively contributes to research in reliability analysis and statistical methodologies.

Mahfooz Alam, Department of Mathematics and Statistics, Faculty of Science and Technology, Vishwakarma University, Pune-411 048, India

Mahfooz Alam recently joined Vishwakarma University as assistant professor in the department of Mathematics and Statistics. He is currently working on moment properties in ordered random variates such as order statistics, record values, progressive censoring, generalized order statistics, dual generalized order statistics and the characterizations of continuous probability distributions. He has published several research papers in the reputated journal of national and international.

Aafaq A. Rather, Symbiosis Statistical Institute, Symbiosis International (Deemed University), Pune, India

Aafaq A. Rather is an Assistant Professor at the Symbiosis Statistical Institute, India. He has published more than 130 research papers in SCI, ESCI, Scopus-indexed, and other reputed international journals. His research interests include probability distributions, reliability analysis, survival analysis and statistical modeling. Dr. Rather is actively involved in research supervision and has contributed significantly to the field of applied statistics through his academic and research activities.

Mehdi Hosseinzadeh, Department of AI, School of Computer Science and Engineering, Galgotias University, Greater Noida, India, International Center for Materials Sciences and Technology, Western Caspian University, Baku, Azerbaijan

Mehdi Hosseinzadeh is the Director of the DTU AI & Data Science Hub (DAIDASH) at Duy Tan University. He has authored over 500 peer-reviewed publications and supervised more than 120 postgraduate researchers, with an h-index above 95. His research focuses on artificial intelligence, deep learning, health informatics, recommender systems, IoT, and data analytics. He has been recognized among the world’s top 2% scientists from 2022 to 2025.

Showkat Ahmad Lone, Department of Basic Science, College of Science and Theoretical Studies, Saudi Electronic University, Riyadh 11673, Kingdom of Saudi Arabia

Showkat Ahmad Lone is an Associate Professor of Mathematics and Statistics at Saudi Electronic University (SEU), Kingdom of Saudi Arabia. He earned his M.Sc. in Statistics from the University of Kashmir in 2013 and his Ph.D. in Statistics, specializing in Reliability Engineering, from Aligarh Muslim University, India, in 2017. His research interests include Statistics, Reliability Engineering, and applied statistical modeling, and he has published extensively in reputed international journals. He actively contributes to research in reliability analysis and statistical methodologies and serves as a reviewer for several leading scientific journals. In recognition of his outstanding research impact and scholarly contributions, he was recently included in Stanford University’s Top 2% Scientists list.

References

Ahsanullah, M. (1995). Record Statistics. Nova Science Publishers, New York.

Ahsanullah, M. and Nevzorov, V. B. (2015). Record via Probability Theory. Atlantis Press, Paris.

Alam, M., Khan, R. U. and Athar, H. (2022). Lower record values from generalized inverse Weibull distribution. J. Math. Model., 10, 93–106.

Alam, M., Khan, M. A. and Khan, R. U. (2020). Characterization of NH distribution through generalized record values. App. Math. E-Notes, 20, 406–414.

Alam, M., Khan, M. A. and Khan, R. U. (2021). On upper k-record values from the generalized linear exponential distribution. J. Stat. Theory Appl., 20, 289–303.

Alam, M., and Vidovic, Z. (2023). K-th Record Moments and Characterization of Inverted Nadarajag-Haghighi Distribution. Pakistan J. Statist. 39, 99–113.

Alam, M., Barakat, H. M., Bakouch, H. S., and Chesneau, C. (2024). Order statistics and record values moments from the Topp-Leone Lomax distribution with applications to entropy. Wireless Personal Communications, 1–19.

Almuhayfith, F. E., Alam, M., Bakouch, H. S., Bapat, S. R., and Albalawi, O. (2024). Linear Combination of Order Statistics Moments from Log-Extended Exponential Geometric Distribution with Applications to Entropy. Mathematics, 12(11), 1744.

Arnold, B. C., Balakrishnan, N. and Nagaraja, H. N. (2008). A First Course in Order Statistics, SIAM Publishers, Philadelphia, USA.

Alsubie, A., Akhter, Z., Athar, H., Alam, M., Ahmad, A. E. B. A., Cordeiro, G. M., & Afify, A. Z. (2021). On the omega distribution: Some properties and estimation. Mathematics, 9(6), 656.

Arnold, B. C., Balakrishnan, N. and Nagaraja, H. N. (2011). Records. New York Wiley.

Balakrishnan, N. and Cohen, A. C. (1992). Order statistics and inference estimation methods, J. R. Stat. Soc. A, 155, 307.

Balakrishnan, N., Malik, H.J. and Ahmed, S.E. (1988): Recurrence relations and identities for moments of order statistics II: Specific continuous distributions, Comm. Statist. Theory Methods 17(8) 2657–2694.

Balakrishnan, N. and Sultan, K.S. (1998): Recurrence relations and identities for moments of order statistics, In: N. Balakrishnan and C. R. Rao, eds., Handbook of Statistics, vol.16, 149-228, Order Statistics: Theory and Methods, North-Holland, Amsterdam, The Netherlands.

Birbiçer, İ. and Genç, A. İ. (2022). On parameter estimation of the standard Omega distribution. Journal of Applied Statistics, 50(15), 3108–-3124. https://doi.org/10.1080/02664763.2022.2101045.

Chandler, K. N. (1952): The distribution and frequency of record values, J. Roy. Statist. Soc. Ser. B, 14, 220–228.

Çetinkaya, Ç., & Genç, A. İ. (2018). Moments of order statistics of the standard two-sided power distribution. Communications in Statistics-Theory and Methods, 47(17), 4311–4328.

Comtet, L. (1974). Advanced Combinatiories. Boston: D. Riedel Publishing Company.

David, H. A. and Nagaraja, H. N. (2003). Order Statistics. 3rd Edn. John Wiley, New York.

Dombi, J., Jónás, T., Tóth, E.Z. and Árva, G. (2019): The Omega probability distribution and its applications in reliability theory. Qual. Reliab. Engng. Int., 35(2), 600–626.

Dwight Chenna, Y. N., Ghassemi, P., Pfefer, T. J., Casamento, J., and Wang, Q. (2018). Free-form deformation approach for registration of visible and infrared facial images in fever screening. Sensors, 18(1), 125. https://doi.org/10.3390/s18010125.

Ghassemi, P., Pfefer, T. J., Casamento, J. P., Simpson, R., and Wang, Q. (2018). Best practices for standardized performance testing of infrared thermographs intended for fever screening. PLOS ONE, 13(9), e0203302. https://doi.org/10.1371/journal.pone.0203302.

Ghitany, M. E., Mazucheli, J., Menezes, A. F. B., and Alqallaf, F. (2019). The unit-inverse Gaussian distribution: A new alternative to two-parameter distributions on the unit interval. Communications in Statistics-Theory and methods, 48(14), 3423–3438.

Gradshteyn, I.S. and Ryzhik, I.M. (2007): Tables of Integrals, Series and Products. Edited by Jeffrey, A. and Zwillinger, D. 7th Ed., Academic Press, San Diego.

Hosking, J. R. M., and Wallis, J. R. (1997). Regional frequency analysis (p. 240).

Hosking, J. R. M. (1990). L-moments: analysis and estimation of distributions using linear combinations of order statistics. J. Roy. Statist. Soc. B, 52(1), 105–124.

Johnson, N. L., Kotz, S., and Balakrishnan, N. (1995). Continuous univariate distributions, volume 2. John wiley & sons.

Jónás, T., and Bakouch, H. S. (2022). The generalized omega function and its connection with some probability distributions. Mediterranean Journal of Mathematics, 19(6), 250.

Khan, A.H., Yaqub, M. and Parvez, S. (1983a): Recurrence relations between moments of order statistics. Naval Research Logistics Quarterly, 30, 419-441. [Corrections, 32, (1985) 693]

Korkmaz, M. Ç. (2020). The unit generalized half normal distribution: A new bounded distribution with inference and application.

Malik, H.J., Balakrishnan, N. and Ahmed, S.E. (1988): Recurrence relations and identities for moments of order statistics I: Arbitrary continuous distribution, Comm. Statist. Theory Methods 17(8) 2632–2655.

Martinez-Florez, G., Olmos, N. M., and Venegas, O. (2024). Unit-bimodal Birnbaum-Saunders distribution with applications. Communications in Statistics-Simulation and Computation, 53(5), 2173– 2192.

Mathai, A. M. and Saxena, R. K. (1973). Generalized hyper-geometric functions with applications in statistics and physical science. Lecture Notes in Mathematics. 348, Springer-Veriag, Berlin.

Maya, R., Jodra, P., Irshad, M. R., and Krishna, A. (2024). The unit Muth distribution: Statistical properties and applications. Ricerche di Matematica, 73(4), 1843–1866.

Mazucheli, J., A. F. B. Menezes, and M. E. Ghitany. “The unit-Weibull distribution and associated inference.” J. Appl. Probab. Stat 13.2 (2018): 1–22.

Mazucheli, J., Menezes, A. F. B., and Chakraborty, S. (2019). On the one parameter unit-Lindley distribution and its associated regression model for proportion data. Journal of Applied Statistics, 46(4), 700–714.

Nadarajah, S. (2008). Explicit expressions for moments of order statistics. Statistics & Probability Letters, 78(2), 196–205.

Nagaraja, H. N. (2013). Moments of order statistics and L-moments for the symmetric triangular distribution. Statistics & Probability Letters, 83(10), 2357–2363.

Okorie, I. E., and Nadarajah, S. (2019). On the omega probability distribution. Quality and Reliability Engineering International, 35(6), 2045–2050.

Özbilen, Ö., and Genç, A. İ. (2022). A bivariate extension of the Omega distribution for two-dimensional proportional data. Mathematica Slovaca, 72(6).

Prataviera, F. and Cordeiro, G. M. (2024). The Unit Omega Distribution, Properties and Its Application. American Journal of Mathematical and Management Sciences, 1–-14. https://doi.org/10.1080/01966324.2024.2310648.

Ribeiro-Reis, L. D. (2021). Unit log-logistic distribution and unit log-logistic regression model. Journal of the Indian Society for Probability and Statistics, 22(2), 375–388.

Singh, B., Alam, I., Rather, A. A., and Alam, M. (2023). Linear combination of order statistics of exponentiated Nadarajah–Haghighi distribution and their applications. Lobachevskii Journal of Mathematics, 44(11), 4839–4848.

Wang, Q., Zhou, Y., Ghassemi, P., McBride, D., Casamento, J. and Pfefer, T., (2022): Infrared Thermography for Measuring Elevated Body Temperature: Clinical Accuracy, Calibration, and Evaluation. Sensors, 22, 215. https://doi.org/10.3390/s22010215.

Yousaf, F., Ali, S., and Shah, I. (2019). Statistical inference for the Chen distribution based on upper record values. Annals of Data Science, 6(4), 831–851.

Yousaf, F., Ali, S., Shah, I., and Riaz, S. (2023). Parameters estimation of the exponentiated chen distribution based on upper record values. Journal of Reliability and Statistical Studies, 197–228.

Zhou, Y., Ghassemi, P., Chen, M., McBride, D., Casamento, J. P., Pfefer, T. J., and Wang, Q. (2020). Clinical evaluation of fever-screening thermography: Impact of consensus guidelines and facial measurement location. Journal of Biomedical Optics, 25(9), 097002. https://doi.org/10.1117/1.JBO.25.9.097002.

Downloads

Published

2026-08-07

How to Cite

Alkhathami, A. A. ., Alam, M. ., Rather, A. A. ., Hosseinzadeh, M. ., & Lone, S. A. . (2026). The Unit Omega Distribution as an Alternative to the Beta Distribution for the Modeling to the Infrared Thermography Temperature Data. Journal of Reliability and Statistical Studies, 19(02), 473–498. https://doi.org/10.13052/jrss0974-8024.19210

Issue

2026: Vol 19 Iss 2

Section

Articles