Johnson SU distribution

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Johnson's SU
Probability density function
JohnsonSU
Cumulative distribution function
Johnson SU
Parameters γ , ξ , δ > 0 , λ > 0 {\displaystyle \gamma ,\xi ,\delta >0,\lambda >0} {\displaystyle \gamma ,\xi ,\delta >0,\lambda >0} (real)
Support − ∞  to  + ∞ {\displaystyle -\infty {\text{ to }}+\infty } {\displaystyle -\infty {\text{ to }}+\infty }
PDF δ λ 2 π 1 1 + ( x − ξ λ ) 2 e − 1 2 ( γ + δ sinh − 1 ⁡ ( x − ξ λ ) ) 2 {\displaystyle {\frac {\delta }{\lambda {\sqrt {2\pi }}}}{\frac {1}{\sqrt {1+\left({\frac {x-\xi }{\lambda }}\right)^{2}}}}e^{-{\frac {1}{2}}\left(\gamma +\delta \sinh ^{-1}\left({\frac {x-\xi }{\lambda }}\right)\right)^{2}}} {\displaystyle {\frac {\delta }{\lambda {\sqrt {2\pi }}}}{\frac {1}{\sqrt {1+\left({\frac {x-\xi }{\lambda }}\right)^{2}}}}e^{-{\frac {1}{2}}\left(\gamma +\delta \sinh ^{-1}\left({\frac {x-\xi }{\lambda }}\right)\right)^{2}}}
CDF Φ ( γ + δ sinh − 1 ⁡ ( x − ξ λ ) ) {\displaystyle \Phi \left(\gamma +\delta \sinh ^{-1}\left({\frac {x-\xi }{\lambda }}\right)\right)} {\displaystyle \Phi \left(\gamma +\delta \sinh ^{-1}\left({\frac {x-\xi }{\lambda }}\right)\right)}
Mean ξ − λ exp ⁡ δ − 2 2 sinh ⁡ ( γ δ ) {\displaystyle \xi -\lambda \exp {\frac {\delta ^{-2}}{2}}\sinh \left({\frac {\gamma }{\delta }}\right)} {\displaystyle \xi -\lambda \exp {\frac {\delta ^{-2}}{2}}\sinh \left({\frac {\gamma }{\delta }}\right)}
Median ξ + λ sinh ⁡ ( − γ δ ) {\displaystyle \xi +\lambda \sinh \left(-{\frac {\gamma }{\delta }}\right)} {\displaystyle \xi +\lambda \sinh \left(-{\frac {\gamma }{\delta }}\right)}
Variance λ 2 2 ( exp ⁡ ( δ − 2 ) − 1 ) ( exp ⁡ ( δ − 2 ) cosh ⁡ ( 2 γ δ ) + 1 ) {\displaystyle {\frac {\lambda ^{2}}{2}}(\exp(\delta ^{-2})-1)\left(\exp(\delta ^{-2})\cosh \left({\frac {2\gamma }{\delta }}\right)+1\right)} {\displaystyle {\frac {\lambda ^{2}}{2}}(\exp(\delta ^{-2})-1)\left(\exp(\delta ^{-2})\cosh \left({\frac {2\gamma }{\delta }}\right)+1\right)}
Skewness − λ 3 e δ − 2 ( e δ − 2 − 1 ) 2 ( ( e δ − 2 ) ( e δ − 2 + 2 ) sinh ⁡ ( 3 γ δ ) + 3 sinh ⁡ ( γ δ ) ) 4 ( Variance ⁡ X ) 1.5 {\displaystyle -{\frac {\lambda ^{3}{\sqrt {e^{\delta ^{-2}}}}(e^{\delta ^{-2}}-1)^{2}((e^{\delta ^{-2}})(e^{\delta ^{-2}}+2)\sinh({\frac {3\gamma }{\delta }})+3\sinh({\frac {\gamma }{\delta }}))}{4(\operatorname {Variance} X)^{1.5}}}} {\displaystyle -{\frac {\lambda ^{3}{\sqrt {e^{\delta ^{-2}}}}(e^{\delta ^{-2}}-1)^{2}((e^{\delta ^{-2}})(e^{\delta ^{-2}}+2)\sinh({\frac {3\gamma }{\delta }})+3\sinh({\frac {\gamma }{\delta }}))}{4(\operatorname {Variance} X)^{1.5}}}}
Excess kurtosis λ 4 ( e δ − 2 − 1 ) 2 ( K 1 + K 2 + K 3 ) 8 ( Variance ⁡ X ) 2 {\displaystyle {\frac {\lambda ^{4}(e^{\delta ^{-2}}-1)^{2}(K_{1}+K_{2}+K_{3})}{8(\operatorname {Variance} X)^{2}}}} {\displaystyle {\frac {\lambda ^{4}(e^{\delta ^{-2}}-1)^{2}(K_{1}+K_{2}+K_{3})}{8(\operatorname {Variance} X)^{2}}}}
K 1 = ( e δ − 2 ) 2 ( ( e δ − 2 ) 4 + 2 ( e δ − 2 ) 3 + 3 ( e δ − 2 ) 2 − 3 ) cosh ⁡ ( 4 γ δ ) {\displaystyle K_{1}=\left(e^{\delta ^{-2}}\right)^{2}\left(\left(e^{\delta ^{-2}}\right)^{4}+2\left(e^{\delta ^{-2}}\right)^{3}+3\left(e^{\delta ^{-2}}\right)^{2}-3\right)\cosh \left({\frac {4\gamma }{\delta }}\right)} {\displaystyle K_{1}=\left(e^{\delta ^{-2}}\right)^{2}\left(\left(e^{\delta ^{-2}}\right)^{4}+2\left(e^{\delta ^{-2}}\right)^{3}+3\left(e^{\delta ^{-2}}\right)^{2}-3\right)\cosh \left({\frac {4\gamma }{\delta }}\right)}
K 2 = 4 ( e δ − 2 ) 2 ( ( e δ − 2 ) + 2 ) cosh ⁡ ( 3 γ δ ) {\displaystyle K_{2}=4\left(e^{\delta ^{-2}}\right)^{2}\left(\left(e^{\delta ^{-2}}\right)+2\right)\cosh \left({\frac {3\gamma }{\delta }}\right)} {\displaystyle K_{2}=4\left(e^{\delta ^{-2}}\right)^{2}\left(\left(e^{\delta ^{-2}}\right)+2\right)\cosh \left({\frac {3\gamma }{\delta }}\right)}
K 3 = 3 ( 2 ( e δ − 2 ) + 1 ) {\displaystyle K_{3}=3\left(2\left(e^{\delta ^{-2}}\right)+1\right)} {\displaystyle K_{3}=3\left(2\left(e^{\delta ^{-2}}\right)+1\right)}

The Johnson's SU-distribution is a four-parameter family of probability distributions first investigated by N. L. Johnson in 1949.[1][2] Johnson proposed it as a transformation of the normal distribution:[1]

z = γ + δ sinh − 1 ⁡ ( x − ξ λ ) {\displaystyle z=\gamma +\delta \sinh ^{-1}\left({\frac {x-\xi }{\lambda }}\right)} {\displaystyle z=\gamma +\delta \sinh ^{-1}\left({\frac {x-\xi }{\lambda }}\right)}

where z ∼ N ( 0 , 1 ) {\displaystyle z\sim {\mathcal {N}}(0,1)} {\displaystyle z\sim {\mathcal {N}}(0,1)}.

Generation of random variables

Let U be a random variable that is uniformly distributed on the unit interval [0, 1]. Johnson's SU random variables can be generated from U as follows:

x = λ sinh ⁡ ( Φ − 1 ( U ) − γ δ ) + ξ {\displaystyle x=\lambda \sinh \left({\frac {\Phi ^{-1}(U)-\gamma }{\delta }}\right)+\xi } {\displaystyle x=\lambda \sinh \left({\frac {\Phi ^{-1}(U)-\gamma }{\delta }}\right)+\xi }

where Φ is the cumulative distribution function of the normal distribution.

Johnson's SB-distribution

N. L. Johnson[1] firstly proposes the transformation :

z = γ + δ log ⁡ ( x − ξ ξ + λ − x ) {\displaystyle z=\gamma +\delta \log \left({\frac {x-\xi }{\xi +\lambda -x}}\right)} {\displaystyle z=\gamma +\delta \log \left({\frac {x-\xi }{\xi +\lambda -x}}\right)}

where z ∼ N ( 0 , 1 ) {\displaystyle z\sim {\mathcal {N}}(0,1)} {\displaystyle z\sim {\mathcal {N}}(0,1)}.

Johnson's SB random variables can be generated from U as follows:

y = ( 1 + e − ( z − γ ) / δ ) − 1 {\displaystyle y={\left(1+{e}^{-\left(z-\gamma \right)/\delta }\right)}^{-1}} {\displaystyle y={\left(1+{e}^{-\left(z-\gamma \right)/\delta }\right)}^{-1}}
x = λ y + ξ {\displaystyle x=\lambda y+\xi } {\displaystyle x=\lambda y+\xi }

The SB-distribution is convenient to Platykurtic distributions (Kurtosis). To simulate SU, sample of code for its density and cumulative distribution function is available here

Applications

Johnson's S U {\displaystyle S_{U}} {\displaystyle S_{U}}-distribution has been used successfully to model asset returns for portfolio management.[3] This comes as a superior alternative to using the Normal distribution to model asset returns. An R package, JSUparameters, was developed in 2021 to aid in the estimation of the parameters of the best-fitting Johnson's S U {\displaystyle S_{U}} {\displaystyle S_{U}}-distribution for a given dataset. Johnson distributions are also sometimes used in option pricing, so as to accommodate an observed volatility smile; see Johnson binomial tree.

An alternative to the Johnson system of distributions is the quantile-parameterized distributions (QPDs). QPDs can provide greater shape flexibility than the Johnson system. Instead of fitting moments, QPDs are typically fit to empirical CDF data with linear least squares.

Johnson's S U {\displaystyle S_{U}} {\displaystyle S_{U}}-distribution is also used in the modelling of the invariant mass of some heavy mesons in the field of B-physics.[4]

References

  1. Johnson, N. L. (1949). "Systems of Frequency Curves Generated by Methods of Translation". Biometrika. 36 (1/2): 149–176. doi:10.2307/2332539. JSTOR 2332539.
  2. Johnson, N. L. (1949). "Bivariate Distributions Based on Simple Translation Systems". Biometrika. 36 (3/4): 297–304. doi:10.1093/biomet/36.3-4.297. JSTOR 2332669.
  3. Tsai, Cindy Sin-Yi (2011). "The Real World is Not Normal" (PDF). Morningstar Alternative Investments Observer.
  4. As an example, see: LHCb Collaboration (2022). "Precise determination of the B s 0 {\displaystyle {B}_{\mathrm {s} }^{0}} {\displaystyle {B}_{\mathrm {s} }^{0}}– B ¯ s 0 {\displaystyle {\overline {B}}_{\mathrm {s} }^{0}} {\displaystyle {\overline {B}}_{\mathrm {s} }^{0}} oscillation frequency". Nature Physics. 18: 1–5. arXiv:2104.04421. doi:10.1038/s41567-021-01394-x.

Further reading

  • Hill, I. D.; Hill, R.; Holder, R. L. (1976). "Algorithm AS 99: Fitting Johnson Curves by Moments". Journal of the Royal Statistical Society. Series C (Applied Statistics). 25 (2).
  • Jones, M. C.; Pewsey, A. (2009). "Sinh-arcsinh distributions" (PDF). Biometrika. 96 (4): 761. doi:10.1093/biomet/asp053.( Preprint)
  • Tuenter, Hans J. H. (November 2001). "An algorithm to determine the parameters of SU-curves in the Johnson system of probability distributions by moment matching". The Journal of Statistical Computation and Simulation. 70 (4): 325–347. doi:10.1080/00949650108812126. MR 1872992. Zbl 1098.62523.