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 Frechet 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.
Highlights
The Cauchy distribution, named after Augustin Cauchy, is a simple family of distributions for which the expected value does not exist
We fitted the two data sets to the GC(α, β, θ) distribution and compared the results with Cauchy, gamma-Pareto proposed by Alzaatreh et al (2012) and betaCauchy distributions proposed by Alshawarbeh et al (2013)
A member of the T-Cauchy{Y} family, the gammaCauchy{exponential} distribution, is studied in detail. This distribution is interesting as it consists of exponentiated Cauchy distribution and distributions of record values of Cauchy distribution as special cases
Summary
The Cauchy distribution, named after Augustin Cauchy, is a simple family of distributions for which the expected value does not exist. 2. The T-Cauchy{Y} family of distributions The T-R{Y} framework defined in Aljarrah et al (2014) (see Alzaatreh et al 2014) is given as follows. Let R be a random variable that follows the Cauchy distribution with PDF fR(x) = fC(x) = π− 1θ− 1(1 + (x/θ)2)− 1 and CDF FR(x) = FC(x) = 0.5 + π− 1 tan− 1(x/θ), x ∈ R, θ > 0, (3) reduces to f ðF C ðxÞÞÞ ðF C ðxÞÞÞ
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