The generalized Cauchy family of distributions with applications

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• Ayman AlzaatrehEmail author,
• Carl Lee,
• Felix Famoye and
• Indranil Ghosh
  

Journal of Statistical Distributions and Applications20163:12

DOI: 10.1186/s40488-016-0050-3

©  The Author(s). 2016

[size=0.8125]Received: 23 February 2016

[size=0.8125]Accepted: 13 July 2016

[size=0.8125]Published: 3 August 2016

Abstract

A family of generalized Cauchy distributions, T-Cauchy{Y} family, is proposed using the T-R{Y} framework. The family of distributions is generated using the quantile functions of uniform, exponential, log-logistic, logistic, extreme value, and Fréchet distributions. Several general properties of the T-Cauchy{Y} family are studied in detail including moments, mean deviations and Shannon’s entropy. Some members of the T-Cauchy{Y} family are developed and one member, gamma-Cauchy{exponential} distribution, is studied in detail. The distributions in the T-Cauchy{Y} family are very flexible due to their various shapes. The distributions can be symmetric, skewed to the right or skewed to the left.

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关键词：distribution Applications Application Generalized Generalize family

The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) function, or Breit–Wigner distribution.
The Cauchy distribution is often used in statistics as the canonical example of a "pathological" distribution since both its mean and its variance are undefined. (But see the section Explanation of undefined moments below.) The Cauchy distribution does not have finite moments of order greater than or equal to one; only fractional absolute moments exist.[1] The Cauchy distribution has no moment generating function.

The Cauchy distribution {\displaystyle f(x;x_{0},\gamma )} f(x;x_{0},\gamma ) is the distribution of the X-intercept of a ray issuing from {\displaystyle (x_{0},\gamma )} (x_{0},\gamma ) with a uniformly distributed angle. Its importance in physics is the result of it being the solution to the differential equation describing forced resonance.[2] In mathematics, it is closely related to the Poisson kernel, which is the fundamental solution for the Laplace equation in the upper half-plane. In spectroscopy, it is the description of the shape of spectral lines which are subject to homogeneous broadening in which all atoms interact in the same way with the frequency range contained in the line shape. Many mechanisms cause homogeneous broadening, most notably collision broadening.[3]
It is one of the few distributions that is stable and has a probability density function that can be expressed analytically, the others being the normal distribution and the Lévy distribution.

Functions with the form of the Cauchy distribution were studied by mathematicians in the 17th century, but in a different context and under the title of the Witch of Agnesi. Despite its name, the first explicit analysis of the properties of the Cauchy distribution was published by the French mathematician Poisson in 1824, with Cauchy only becoming associated with it during an academic controversy in 1853. As such, the name of the distribution is a case ofStigler's Law of Eponymy. Poisson noted that if the mean of observations following such a distribution were taken, the mean error did not converge to any finite number. As such,Laplace's use of the Central Limit Theorem with such a distribution was inappropriate, as it assumed a finite mean and variance. Despite this, Poisson did not regard the issue as important, in contrast to Bienaymé, who was to engage Cauchy in a long dispute over the matter.

The Cauchy distribution, named after Augustin Cauchy, is a simple family of distributions for which the expected value does not exist. Also, the family is closed under the formation of sums of independent random variables, and hence is an infinitely divisible family of distributions. The Cauchy distribution was used by Stigler (1989) to obtain an explicit expression for P(Z 1 ≤ 0, Z 2 ≤ 0) where (Z 1, Z 2) Tfollows the standard bivariate normal distribution. The Cauchy distribution has been used in many applications such as mechanical and electrical theory, physical anthropology, measurement problems, risk and financial analysis. It was also used to model the points of impact of a fixed straight line of particles emitted from a point source (Johnson et al. 1994). In Physics, it is called a Lorenzian distribution, where it is the distribution of the energy of an unstable state in quantum mechanics.

Eugene et al. (2002) introduced the beta-generated family of distributions using the beta as the baseline distribution.

Based on the beta-generated family, Alshawarbeh et al. (2013) proposed the beta-Cauchy distribution.