Johnson's SU-distribution

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:[3]

Johnson's SU
Probability density function
Cumulative distribution function
Parameters (real)
Support
PDF
CDF
Mean
Median
Variance

where .

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:

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

Johnson's SB-distribution

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

where .

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

where Φ is the cumulative distribution function of the normal distribution. SB-distribution is convenient to Platykurtic distributions (Kurtosis).

Applications

Johnson's -distribution has been used successfully to model asset returns for portfolio management.[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. Johnson (1949) "Systems of Frequency Curves...", p. 158
  4. Tsai, Cindy Sin-Yi (2011). "The Real World is Not Normal" (PDF). Morningstar Alternative Investments Observer.

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.
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