Abstract
In this paper, a method for generating a new family of univariate continuous distributions using the tangent function is proposed. Some general properties of this new family are discussed: hazard function, quantile function, Renyi and Shannon entropies, symmetry, and existence of the non-central n^th moment. Some new members as sub-families in the T-X family of distributions are provided. Three members of the new sub-families are defined and discussed: the five-parameter Normal-Generalized hyperbolic secant distribution (NGHS), the five-parameter Gumbel-Generalized hyperbolic secant distribution (GGHS), and the six-parameter Generalized Error-Generalized hyperbolic secant distribution (GEHS), the shapes of these distributions were found: skewed right, skewed left, or symmetric, and unimodal, bimodal, or trimodal. Finally, to demonstrate the usefulness and the capability of the distributions, two real data sets are used and the results are compared with other known distributions.
Highlights
Statistical distribution is a mathematical description of a random phenomenon in terms of the probabilities of events
Proof: By equation (3.3) in Lemma 1 above, since the quantile of the random variable Xf with cumulative distribution function (CDF) F(x) is the quantile of the Generalized Hyperbolic Secant distribution with parameters α and β, and it is given by F−1(λ; α, β) = QXf(λ; α, β) = (π/ 2)β sinh−1{tan[π(λ − 1/2)]} + α, since tan−1(y) = cot−1(−y) − π/2, it can be gotten the result in (3.7)
Proof: By equation (3.19) in Lemma 12 above, since the quantile of the random variable Xf with CDF F(x) is the quantile of the Generalized Hyperbolic Secant with parameters α and β, and it is given by F−1(λ; α, β) = QXf(λ; α, β) = (π/2)β sinh−1{tan[π(λ − 1/2)]} + α
Summary
Statistical distribution is a mathematical description of a random phenomenon in terms of the probabilities of events. A more general class from classes in (1.2), (1.3) and (1.4) has been introduced by Alzaatreh et al (2013a) This new class is depending on replacing the beta PDF b(t) with a PDF r(t) of a continuous random variable T ∈ [a, b], −∞ ≤ a < b ≤ ∞, and the CDF F(x) with a function W(F(x)), where W(∙) satisfies the following conditions: i. These R codes are available to the reader from the author
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