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. strangely, 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 adequate cognizance within the released literature and will be necessary for college students and practitioners.” over the last few years, even if, numerous generalizations of the hyperbolic secant distribution became renowned within the context of monetary go back information as a result of its very good healthy. the majority of them are summarized inside of this Springer Brief.

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

**Example text**

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. : The generalized secant hyperbolic family and its properties. Commun. Stat. Theory Methods 31(2), 219–238 (2002) 2. : Classes of skewed generalized hyperbolic secant distributions. Working Paper No. 45 (unpublished). Department of Statistics and Econometrics, FAU Erlangen-Nuremberg, Nuremberg (2002) 3.

1) where g denotes the transformed density, f and F the (symmetric) density and cumulative distribution function, respectively, of the original (“parent”) distribution and w is an appropriate weighting function on the interval (0, 1) with parameter vector π M. J. 1007/978-3-642-45138-6_4, © The Author(s) 2014 37 38 4 The BHS Distribution Family (see, for instance, Ferreira and Steel [8]). Choosing w(u; φ) = 2F(φF −1 (u)), the skewing mechanism of Azzalini [2, 3] is recovered. 2) corresponds to applying different parameters of scale to the positive and the negative part of a symmetric density (see, for example, Fernández et al.

256). The standardized counterpart (see Fig. 1) with variance one is ∈ 2ρ−2 ρ · Γ f (x; ρ) √ π Γ (ρ) ∞ ∈ ρx ρ +i 2 2 2 . 3) Simpler forms of the density are provided by Baten [3] and Harkness and Harkness [11] for ρ ∈ N. For ρ = 1, the hyperbolic secant density from Chap. 5 + ix)|2 = π . 2), the corresponding counterpart is |Γ (n + ix)|2 = π Pn for n = 1, 2, . . and Pn √ x sinh(π x) n (i − 1)2 + x 2 . 2 GHS Distribution: Properties 1. , using suitable integration algorithms). In addition, a rough approximation is available based on an Edgeworth expansion (see Harkness and Harkness [11]) 1 ∈ ∈ Φ (3) (x/ ρ).