Hyperbolic Cosine Rayleigh Distribution and Its Application to Breaking Stress of Carbon Fibers
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RESEARCH ARTICLE
Hyperbolic Cosine Rayleigh Distribution and Its Application to Breaking Stress of Carbon Fibers K. M. Sakthivel1
•
J. Rajkumar1
Accepted: 18 September 2020 Ó The Indian Society for Probability and Statistics (ISPS) 2020
Abstract In this paper, we introduce generalization of Rayleigh distribution using hyperbolic cosine family distribution and it is named as hyperbolic cosine Rayleigh distribution. Further we present various mathematical properties of the proposed distribution including explicit expressions for moments, quantiles, asymptotic behaviour, order statistics, stochastic orderings and reliability function. Maximum likelihood estimation method is used to obtain estimators for the parameters of the distribution. Finally we have shown that the proposed model has application in studying breaking stress of carbon fibers. Keywords Hyperbolic cosine family distribution Rayleigh distribution Reliability analysis Maximum likelihood estimation
1 Introduction Life time distributions are inevitable in several areas such as engineering, actuarial science, environmental, biological studies, economics, finance and insurance. Several lifetime distributions have been used to model life time data. The quality of the procedures used in a statistical analysis depends heavily on the assumed probability model or distribution. Because of this, considerable effort has been expended in the development of large classes of standard probability distributions along with relevant statistical methodologies. Several methods for generating new families of distributions have been studied. Recently Kharazmi and Saadatinik (2016) introduced a family of distributions using Hyperbolic Cosine-F (HCF) family and Kharazmi et al. (2019) proposed a new continuous lifetime distribution and some of its mathematical properties with application to the indemnity and aircraft windshield datasets. This new class of probability & K. M. Sakthivel [email protected] 1
Department of Statistics, Bharathiar University, Coimbatore, Tamil Nadu 641046, India
123
Journal of the Indian Society for Probability and Statistics
distribution is obtained by compounding a baseline F distribution with the hyperbolic cosine function. This technique resulted in adding an extra parameter to a family of distribution for more flexibility. The hyperbolic cosine has similar name to the trigonometric functions, but it is defined in terms of the exponential function as follows coshð xÞ ¼
ex þ ex 2
ð1Þ
The function coshð xÞ is even and has a Taylor series expression with only even exponents for x as follows coshð xÞ ¼
1 X x2n ð2nÞ! n¼0
ð2Þ
Let X be a continuous random variable with cumulative distribution function (CDF) F ð xÞ then the model is called hyperbolic cosine-F (HCF) distribution and its probability density function (PDF) is given as gð x; aÞ ¼
2aea f ð xÞ coshðaF ð xÞÞ; x [ 0; a [ 0 e2a 1
ð3Þ
The motivation of the HCF distribution is given as follows Representation 1 Suppose the failure of a device occurs due to the presen
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