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Some New Facts about the Unit-Rayleigh Distribution with Applications

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Department of Marine Geology, Faculty of Marine Science, King Abdulaziz University, Jeddah 21551, Saudi Arabia
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LMNO, Université de Caen Normandie, Campus II, Science 3, 14032 Caen, France
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Department of Statistics, The Islamia University of Bahawalpur, Punjab 63100, Pakistan
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The Higher Institute of Commercial Sciences, Al mahalla Al kubra, Algarbia 31951, Egypt
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Department of Computer Science and IT, GLIM Institute of Modern Studies Bahawalpur, Bahawalpur, Punjab 63100, Pakistan
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Department of Computer Science, Govt Degree College for Women Ahmadpur East, Bahawalpur 63350, Pakistan
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Author to whom correspondence should be addressed.
Mathematics 2020, 8(11), 1954; https://doi.org/10.3390/math8111954
Received: 16 September 2020 / Revised: 26 October 2020 / Accepted: 28 October 2020 / Published: 4 November 2020
The unit-Rayleigh distribution is a one-parameter distribution with support on the unit interval. It is defined as the so-called unit-Weibull distribution with a shape parameter equal to two. As a particular case among others, it seems that it has not been given special attention. This paper shows that the unit-Rayleigh distribution is much more interesting than it might at first glance, revealing closed-form expressions of important functions, and new desirable properties for application purposes. More precisely, on the theoretical level, we contribute to the following aspects: (i) we bring new characteristics on the form analysis of its main probabilistic and reliability functions, and show that the possible mode has a simple analytical expression, (ii) we prove new stochastic ordering results, (iii) we expose closed-form expressions of the incomplete and probability weighted moments at the basis of various probability functions and measures, (iv) we investigate distributional properties of the order statistics, (v) we show that the reliability coefficient can have a simple ratio expression, (vi) we provide a tractable expansion for the Tsallis entropy and (vii) we propose some bivariate unit-Rayleigh distributions. On a practical level, we show that the maximum likelihood estimate has a quite simple closed-form. Three data sets are analyzed and adjusted, revealing that the unit-Rayleigh distribution can be a better alternative to standard one-parameter unit distributions, such as the one-parameter Kumaraswamy, Topp–Leone, one-parameter beta, power and transmuted distributions. View Full-Text
Keywords: unit-Rayleigh distribution; hazard rate function; incomplete moments; order statistics; estimation unit-Rayleigh distribution; hazard rate function; incomplete moments; order statistics; estimation
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MDPI and ACS Style

Bantan, R.A.R.; Chesneau, C.; Jamal, F.; Elgarhy, M.; Tahir, M.H.; Ali, A.; Zubair, M.; Anam, S. Some New Facts about the Unit-Rayleigh Distribution with Applications. Mathematics 2020, 8, 1954. https://doi.org/10.3390/math8111954

AMA Style

Bantan RAR, Chesneau C, Jamal F, Elgarhy M, Tahir MH, Ali A, Zubair M, Anam S. Some New Facts about the Unit-Rayleigh Distribution with Applications. Mathematics. 2020; 8(11):1954. https://doi.org/10.3390/math8111954

Chicago/Turabian Style

Bantan, Rashad A.R., Christophe Chesneau, Farrukh Jamal, Mohammed Elgarhy, Muhammad H. Tahir, Aqib Ali, Muhammad Zubair, and Sania Anam. 2020. "Some New Facts about the Unit-Rayleigh Distribution with Applications" Mathematics 8, no. 11: 1954. https://doi.org/10.3390/math8111954

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