Some New Facts about the Unit-Rayleigh Distribution with Applications

Abstract

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.

On the Arcsecant Hyperbolic Normal Distribution. Properties, Quantile Regression Modeling and Applications

“…Although it has very flexible forms for data modeling, sometimes it is not sufficient for modeling and explaining unit datasets. For this reason, new alternative unit models have been proposed in the statistical distribution literature, including the Johnson S B [1], Topp-Leone [2], Kumaraswamy [3], standart two-sided power [4], log-Lindley [5], log-xgamma [6], unit Birnbaum-Saunders [7], unit Weibull [8], unit Lindley [9], unit inverse Gaussian [10], unit Gompertz [11], second degree unit Lindley [12], log-weighted exponential [13], logit slash [14], unit generalized half normal [15], unit Johnson S U [16], trapezoidal beta [17] and unit Rayleigh [18] distributions. Many of the above distributions were obtained by transforming the baseline distribution, and they performed better than the beta distribution in terms of data modeling.…”

Theory and Applications of the Unit Gamma/Gompertz Distribution

“…As a main statistical work, we analyze this data set via a fitting approach. In this regard, we compare the fit of the proposed UG/G model and those of the Kumaraswamy (Kw), beta (Beta), UR (recalling that it refers to the unit Rayleigh distribution by [10]), Topp-Leone (Topp), power (Power), and Transmuted (TM) models. These competitive models are defined by specific cdfs, which are recalled below.…”

“…This particular scheme is also known to transpose some characteristics of the distribution of X to the unit interval. With this inverse-exponential scheme, we may refer the reader to the construction of the log-gamma distribution developed by [2] and based on the standard gamma distribution, log-Lindley distribution established by [3] and based on the Lindley distribution, unit Weibull distribution examined by [4,5] and using the Weibull distribution, unit Gompertz distribution developed by [6] and based on the Gompertz distribution, unit Burr-XII (UBXII) created by [7] distribution and based on the Burr-XII distribution, log-weighted exponential distribution motivated by [8] and based on the weighted exponential distribution, log-Xgamma distribution proposed by [9] and based on the Xgamma distribution, and unit Rayleigh (UR) distribution studied in [10] and using the Rayleigh distribution, among others.…”

A New Truncated Muth Generated Family of Distributions with Applications

“…Modern applications are numerous in the fields of psychology, economics, biology, and engineering. For more information on this topic, see [21][22][23][24] as well as the references therein. is section is dedicated to the unit truncated M distribution, which will be the main ingredient to the NTM-G family.…”