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Hyperbolic Secant distribution

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The Hyperbolic Secant distribution is symmetric on the real line with density proportional to sech. It has heavier tails than normal while retaining finite moments of every order.

Feature Reading
Support mu is the centre and sigma the scale. It is symmetric and unimodal; its higher-than-normal kurtosis can suit errors with moderate tails.
Shape It resembles logistic through symmetry and tails but is not the same family. Its characteristic function has a compact hyperbolic-cosine form, linking it to transforms and random processes.

Origin and terminology

The hyperbolic secant appeared in work on error curves and time series, and was organized as a statistical distribution through its relation to hyperbolic functions.

In real models

  • symmetric errors with more extremes than normal
  • time-series models and densities arising from hyperbolic transformations

Comparisons

Tables may normalize scale differently; compare sigma only after checking the convention.

A density that reappears in its own transform

The Hyperbolic Secant distribution has an unusual mathematical symmetry: under suitable scaling, its Fourier transform retains the same functional form. This self-duality makes it interesting in harmonic analysis and time-series models.

Its bell is sharper and its tails heavier than the Normal, yet all moments exist. It provides an alternative between Gaussian errors and far more extreme tails such as Cauchy. Visual resemblance to Logistic does not imply equality; their characteristic functions and scale constants differ. Since both are symmetric, clear skewness calls for another family before tail thickness is compared.

Decision guide

A good candidate when: a smooth symmetric law with exponential tails somewhat heavier than Normal is required.

Compare it with: Logistic, Laplace, and Student’s t. Distinguish peak shape and tail decay with Q–Q plots; shared symmetry does not imply equal extreme risk.

References

Hyperbolic Secant Distribution: equations and calculator

Distribution defintion

X∼HyperbolicSecant(μ,σ)X\sim\mathrm{HyperbolicSecant}\left(\mu,\sigma\right)

Distribution domain

x∈(−∞,∞)x\in\left(-\infty,\infty\right)

Parameters domain and parameters constraints

μ∈R,σ∈R+\mu\in\mathbb{R},\sigma\in\mathbb{R}^{+}

Cumulative distribution function

FX(x)=2πarctan⁡[exp⁡ ⁣(π2 z(x))]F_{X}\left(x\right)=\frac{2}{\pi} \arctan\left[\exp\!\left(\frac{\pi}{2}\,z(x)\right)\right]

Probability density function

fX(x)=12σsech⁡ ⁣(π2 z(x))f_{X}\left(x\right)=\frac{1}{2\sigma} \operatorname{sech}\!\left(\frac{\pi}{2}\,z(x)\right)

Percent point function/Sample

FX−1(u)=μ+σ2π ln⁡ ⁣[tan⁡(π2 u)]F^{-1}_{X}\left(u\right)=\mu+\sigma\frac{2}{\pi}\,\ln\!\left[\tan\left(\frac{\pi}{2}\,u\right)\right]

Non-central parametric moments

μ~k′=E[X~k]=∫−∞∞xkfX~(x)dx=1+(−1)k2π22kk![ζ(k+1,14)−ζ(k+1,34)]\tilde{\mu}'_{k}=E[\tilde{X}^k]=\int_{-\infty}^{\infty}x^{k}f_{\tilde{X}}\left(x\right)dx=\frac{1+\left(-1\right)^{k}}{2\pi2^{2k}}k!\left[\zeta\left(k+1,\frac{1}{4}\right)-\zeta\left(k+1,\frac{3}{4}\right)\right]

Parametric mean

Mean(X)=μ+σμ~1′=μ\mathrm{Mean}(X)=\mu+\sigma\tilde{\mu}'_{1}=\mu

Parametric variance

Variance(X)=σ2(μ~2′−μ~1′2)=σ2\mathrm{Variance}(X)=\sigma^{2}(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})=\sigma^{2}

Parametric skewness

Skewness(X)=μ~3′−3μ~2′μ~1′+2μ~1′3(μ~2′−μ~1′2)1.5=0\mathrm{Skewness}(X)=\frac{\tilde{\mu}'_{3}-3\tilde{\mu}'_{2}\tilde{\mu}'_{1}+2\tilde{\mu}'^{3}_{1}}{(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})^{1.5}}=0

Parametric kurtosis

Kurtosis(X)=μ~4′−4μ~1′μ~3′+6μ~1′2μ~2′−3μ~1′4(μ~2′−μ~1′2)2=3\mathrm{Kurtosis}(X)=\frac{\tilde{\mu}'_{4}-4\tilde{\mu}'_{1}\tilde{\mu}'_{3}+6\tilde{\mu}'^{2}_{1}\tilde{\mu}'_{2}-3\tilde{\mu}'^{4}_{1}}{(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})^{2}}=3

Parametric median

Median(X)=μ\mathrm{Median}(X)=\mu

Parametric mode

Mode(X)=μ\mathrm{Mode}(X)=\mu

Additional information and definitions

X~∼HyperbolicSecant(0,1)\tilde{X}\sim\mathrm{HyperbolicSecant}\left(0,1\right)
μ:Location parameter\mu:\text{Location parameter}
σ:Scale parameter\sigma:\text{Scale parameter}
z(x)=(x−μ)/σz\left(x\right)=\left(x-\mu\right)/\sigma
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
ζ(a,s):Hurwitz zeta function\zeta(a,s):\text{Hurwitz zeta function}