G-and-H Distribution

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The g-and-h distribution (a combination of the g-distribution and h-distribution*) is used in statistical modeling as an alternative for Ordinary Least Squares. Developed by Tukey in 1977, it is seldom used perhaps because it does not have a closed-form solution for a probability density function (PDF). Turley (n.d.) states that as the distribution is so difficult to compute, it has very little practical use. However, it is sometimes used as a model for a severity distribution.

Calculating the G-and-H Distribution Parameters

The g-and-h distribution is a powerful tool for data analysis, but it can be tricky to put into practice. By utilizing the methods outlined in Dutta and Babbel (2005) or Turley, P.(n.d.), parameters of this special distribution can be computed indirectly – however these techniques involve sophisticated calculations which are detailed further in Cruz et al’s Handbook on Operational Risk Management Analytics .

Some formulas are available for very particular random variables. For example, Chaudhuri and Ghosh offer the following formula for the g-and-h distribution for a univariate normal random variable Yg,h, defined by a Z-transformation:


  • A and B are scale parameters,
  • g and h control skew and kurtosis.

    The g-distribution and h-distribution can both be derived from the above equation:

    • The g-distribution is found when h = 0. It corresponds to a scaled lognormal distribution when g is a constant.
    • The h-distribution is obtained when g = 0.


    Chaudhuri, A. & Ghosh, S. Retrieved July 8, 2017 from: Quantitative Modeling of Operational Risk in Finance and Banking Using Possibility Theory
    Dutta, K.K. and D.F. Babel (2002). Extracting Probabilistic Information fro the Prices of Interested Rate Options: Test of Distributional Assumptions. The Journal of Business 78(3), 841-870.
    Eghwerido, J. et. al. The Alpha Power Marshall-Olkin-G Distribution: Properties, and Applications Retrieved 12/21/2022 from: https://link.springer.com/article/10.1007/s13171-020-00235-y
    Shakti, P. (2022). Ghosh Distribution. Retrieved 12/21/2022 from: https://p-distribution.com/ghosh-distribution/
    Tukey, J. W. (1977). Exploratory Data Analysis. Reading, MA: Addison-Wesley
    Turley, P. Just a few more moments: the g-and-h distribution. Retrieved July 8, 2017 from: https://www.researchgate.net/publication/251947280_Just_a_few_more_moments_the_g-and-h_distribution