< 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 [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 *Y _{gh}*, 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**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 *τ*_{g,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.