G-and-H Distribution / Tukey g-and-h (TGH) family

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What is the g-and-h distribution?

The g-and-h distribution (a combination of the g-distribution and h-distribution) is mostly used in robust statistics. The Tukey g-and-h (TGH) family of parametric distributions includes the g-and-h distribution as a special case.

Developed by Tukey in 1977 [1], the g-and-h distribution is seldom used perhaps because it does not have a closed-form solution for a probability density function (pdf). A closed form solution is an expression for an exact solution given with a finite amount of data [2].

As the distribution doesn’t have a closed-form solution, this makes it difficult to compute, thus is has very little practical use [3]. However, it is sometimes used as a model for a severity distribution. Other members of the TGH family are in more widespread use, such as special cases the normal (Gaussian) distribution and Pareto-like distributions.

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 [4] or Turley, P.[3], parameters of this special distribution can be computed indirectly – however these techniques involve sophisticated calculations — which are detailed in Cruz et al’s Fundamental Aspects of Operational Risk and Insurance Analytics [5] .

Some formulas are available for very particular random variables. For example, Chaudhuri and Ghosh [6] offer the following formula for the g-and-h distribution for a univariate normal random variable Ygh, defined by the following Z-transformation:

Where:

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

Tukey’s g-distribution and h-distribution.

The g-distribution and h-distribution can both be derived from the above formula: Tukey’s g distribution [7]. Tukey’s g-distribution is found when h = 0. It corresponds to a scaled log-normal distribution when g is a constant.

Similarly, Tukey’s h-distribution is obtained when g = 0.

Tukey g-and-h (TGH) family of parametric distributions

A random variable T, obtained by transforming a standard normal random variable Z with a monotonic TGH transformation τg,h, follows a TGH distribution [8]: Monotonic means that the function is always increasing or always decreasing.

The TGH family has two parameters: g and h:

• g controls the skewness,
• h controls the kurtosis, where h ≥ 0.

Special cases of the TGH family include:

• The g-and-h distribution when g = 0 and h = 0. In this case, the g-and-h distribution is a symmetric distribution with a kurtosis of 3.
• The normal (Gaussian) distribution when g = h = 0,
• Pareto-like distributions [9] when g = 0.
• The shifted log-normal distribution [10] when h = 0.

References:

1. Tukey, J. W. (1977). Exploratory Data Analysis. Reading, MA: Addison-Wesley
2. Mark van Hoeij. Closed form solutions. Florida State University.
3. 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
4. Dutta, K.K. and D.F. Babel (2002). Extracting Probabilistic Information for the Prices of Interested Rate Options: Test of Distributional Assumptions. The Journal of Business 78(3), 841-870.
5. Cruz et al. (2008). Fundamental Aspects of Operational Risk and Insurance Analytics: A Handbook of Operational Risk. Wiley.
6. Chaudhuri, A. & Ghosh, S. Retrieved July 8, 2017 from: Quantitative Modeling of Operational Risk in Finance and Banking Using Possibility Theory
7. Jimenez, J. & Arunachalam, V. The Use of the Tukey’s g − h family of distributions to Calculate Value at Risk and Conditional Value at Risk. Journal of Risk, vol. 13, No. 4, summer, 2011. Retrieved July 4, 2023 from: https://pdfs.semanticscholar.org/9a9c/3f9f73a3e7f43e18a25a1261b9f4f6dfb1c0.pdf
8. Yan, Y. & Genton, M. What is the Tukey g-and-h distribution? SignificanceVolume 16, Issue 3 p. 12-13
9. Newman, M. (2017) Power-law distribution. Significance, 14(4), 10– 11.
10. Limpert, E. and Stahel, W. A. (2017) The log-normal distribution. Significance, 14(1), 8– 9.