By Matthias J. Fischer

Among the symmetrical distributions with an enormous area, the most well-liked replacement to the traditional variation is the logistic distribution in addition to the Laplace or the double exponential distribution, which was once first brought in 1774. sometimes, the Cauchy distribution can be used. unusually, the hyperbolic secant distribution has led a charmed existence, even if Manoukian and Nadeau had already said in 1988 that “... the hyperbolic-secant distribution ... has now not got enough awareness within the released literature and should be helpful for college students and practitioners.” over the last few years, in spite of the fact that, a number of generalizations of the hyperbolic secant distribution became renowned within the context of monetary go back info due to its very good healthy. the vast majority of them are summarized inside this Springer Brief.

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**Additional resources for Generalized Hyperbolic Secant Distributions: With Applications to Finance**

**Example text**

7) ω with normalizing constant ⎡ C1 = bc−λ+kω ⎣ ∈ j=0 ⎤−1 Γ (ω + j) k j κ B(λ, k( j + ω) − λ)⎦ , Γ (ω) j! where B(a, b) denotes the Beta function, λ = a/b and κ = h/c. 7). The corresponding cumulative function is given by F(x) = C1 ∈ Γ (ω+ j) k j κ Bτ (u) (λ, k( j + ω) − λ) j=0 j! ∈ Γ (ω+ j) k j κ B(λ, k( j + ω) − λ) j=0 j! for τ (u) √ c exp(bu) + c and where Bu (a, b) denotes the incomplete Beta function. The above conditions on the parameters ensure the densities are positive, and further that the distribution has well-defined moment-generating functions, and hence all moments finite.

1 Excursion: Moment-Ratio Plots Moment-ratio diagrams (MRD) were introduced for Pearson-type distributions by Elderton and Johnson [8] in order to provide a useful visual assessment of skewness and kurtosis. The classical moment ratio plot consists of all possible pairs of third and fourth standardized moments (M3 , M4 ) that can be obtained through different combinations of the shape parameters of the√underlying distributions, provided their existence. , for a given level of kurtosis only a finite range of skewness may be spanned.

The above conditions on the parameters ensure the densities are positive, and further that the distribution has well-defined moment-generating functions, and hence all moments finite. They can be expressed in terms of the Gamma function and its derivatives. For all S-EGSH members there is a unique mode (Fig. 4). 24 2 The GSH Distribution Family and Skew Versions Fig. 4 S-EGSH distribution: Densities with a = 1, C = 4, h = 1, k = 2, w = 1 and b ∈ [1, 2] (left panel), and a = 1, b = 1, C = 4, h = 1, k = 2 and w ∈ [1, 6] (right panel) References 1.