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

< Probability distribution list < G-and-H Distribution

## 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 , 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 . As the distribution doesn’t have a closed-form solution, this makes it difficult to compute, thus is has very little practical use . 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  or Turley, P., 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  .

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 Ygh, defined by the following Z-transformation:

Where:

## 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 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 :

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  when g = 0.
• The shifted log-normal distribution  when h = 0,

## References:

 Mark van Hoeij. Closed form solutions. Florida State University.

 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

 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.

 Cruz et al. (2008). Fundamental Aspects of Operational Risk and Insurance Analytics: A Handbook of Operational Risk. Wiley.

 Chaudhuri, A. & Ghosh, S. Retrieved July 8, 2017 from: Quantitative Modeling of Operational Risk in Finance and Banking Using Possibility Theory

 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

 Yan, Y. & Genton, M. What is the Tukey g-and-h distribution? SignificanceVolume 16, Issue 3 p. 12-13

 Newman, M. (2017) Power-law distributionSignificance, 14(4), 10– 11.

 Limpert, E. and Stahel, W. A. (2017) The log-normal distribution. Significance, 14(1), 8– 9.

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