< List of probability distributions

The term **GIG distribution** usually refers to the *generalized inverse Gaussian distribution* (e.g. Zhang et al [1], Pothula [2]). However, the term GIG distribution is occasionally used to mean the *generalized integer gamma distribution* instead (e.g. Coelho [3]).

**Contents:**

## 1.Generalized Inverse Gaussian Distribution (GIG distribution)

The Generalized Inverse Gaussian Distribution (GIG), denoted GIG(λ, ψ, χ), is a common distribution used in several areas of statistics, including finance, geostatistics, and statistical linguistics. The GIG model is a popular distribution in probability theory for continuous probability distributions due to its versatility and applicability in different areas of study.

The GIG distribution has the property of being able to produce symmetric as well as asymmetric distributions. It can also yield a distribution model that approaches a normal/Gaussian or even a Laplace distribution. While the GIG is typically used in continuous data sets, being able to model such a broad range of distributions is essential as it makes the GIG very versatile.

The distribution does not have a closed for for its cumulative distribution function (CDF). The probability density function (PDF) can be expressed with Bessel functions as [4]

where x > 0 and Kλ is the modified Bessel function.

Parameters must satisfy the following three requirements:

- χ > 0, ψ ≥ 0, when λ < 0,
- χ > 0, ψ > 0, when λ = 0,
- χ ≥ 0, ψ > 0, when λ > 0.

Special cases of the GOG distribution include the inverse normal/Gaussian distribution (for λ = −1/2) and gamma distribution ( for *ψ* = 0) [5].

The moment generating function is

The name generalized inverse Gaussian distribution originates with Ole Barndorff-Nielsen, who came across the largely underdeveloped distribution while studying hyperbolic distributions. Barndorff-Nielsen et al. [6] proved that any GIG distribution with a non-positive power parameter is the distribution of the first hitting time to level 0 for a time-homogeneous diffusion process with state space [0,∞). This suggests that the GIG distribution could be useful as a lifetime distribution or the time distribution between successive events in a renewal process [7].

Although Barndorff-Nielsen popularized the GIG distribution, it was Étienne Halphen who first discovered the GIG distribution in 1941 (albeit under a different name). Its properties are discussed in Bent Jørgensen’s *Lecture Notes in Statistics* [8].

## Applications

GIG distribution has multiple applications, and it has a wide range of uses in many industries. One common application of the GIG distribution is in finance, specifically in the pricing of options. Options are complex financial instruments used to hedge against stock price volatility. Options pricing models that incorporate GIG distribution are preferred due to their ability to handle skewness and kurtosis more efficiently than standard models.

Another application of the GIG distribution is its use in spatial statistics, particularly in the study of geostatistics. Geo-statistics is used to analyze and understand geographically dependent data that has a spatial correlation. By using the GIG distribution, researchers can model the data more accurately, incorporating skewness, and heavy tails in their predictions.

Statistical linguistics is a rapidly growing field, and it also benefits from the use of the GIG distribution. In statistical linguistics, texts are analyzed to understand the underlying language structure best. In addition, model inference can be used to generate texts that are grammatically correct and stylistically similar to the input corpus. The GIG distribution is helpful in these cases, as it can generate syntactically valid texts within the style of the original input.

## 2. Generalized integer gamma distribution

The generalized integer gamma distribution (also called the GIG distribution) is a special case of the generalized chi-squared distribution and involves the distribution of the sum of independent gamma distributed random variables with distinct rate parameters and integer shape parameters. It is also known as the discrete gamma distribution or the integer gamma distribution.

The probability density function (PDF) of the generalized integer gamma distribution if [9]

Where

- p=depth
- r
_{j}= shape parameters - λ
_{j}= rate parameters.

And:

and for k = 1, r_{j – 1}; j = 1, … , p,

where

The integer shape parameter in the distribution is responsible for its discrete nature. In other words, it affects the distribution’s probability mass function (PMF), which only takes integer values. Moreover, this distribution is a special case of the chi-squared distribution where the random variables are gamma distributed.

The generalized integer gamma distribution has several interesting properties:

- It has a mean value of integer shape parameter divided by the rate parameter,
- The variance is the integer shape parameter divided by the square of the rate parameter,
- It has a mode of floor(x), where the variable x is the sum of the gamma-distributed random variables.

The cumulative density function (CDF) of the GIG distribution is expressed as the product of the terms from the gamma distribution formula for the individual variates:

Although the generalized integer gamma distribution is not as commonly used as some other distributions, it has several applications. For instance, it is useful in queuing theory, probability theory, and the analysis of discrete data. Its discrete nature makes it relevant in situations where the data points are discrete, and the underlying distribution is unknown. It is also used in actuarial science to model the number of claims an insurance company receives in a specified period.

When working with the generalized integer gamma distribution, it is essential to note that there is no analytical expression for the probability mass function (PMF) of the sum of gamma-distributed variables. Therefore, the distribution parametrization must be well defined. Additionally, the distribution parameters are only identifiable up to a permutation of the variables, so it is crucial to partition the observations appropriately.

## References

[1] Zhang et. al. EP-GIG Priors and Applications in Bayesian Sparse Learning Journal of Machine Learning Research 13 (2012) 2031-2061 Submitted 9/11; Revised 2/12; Published 6/12

[2] Pothula, P. Random Variate Generation from Generalized Inverse

Gaussian Distribution. Thesis. Retrieved April 20, 2023 from: https://scholarworks.unr.edu/bitstream/handle/11714/4019/Pothula_unr_0139M_10896.pdf?sequence=1&isAllowed=y

[3] Coelho, C. A mixture of Generalized Integer Gamma distributions as the exact distribution of the product of an odd number of independent Beta random variables: applications. Journal of Interdisciplinary Mathematics Vol. 9 (2006), No. 2, pp. 229–248. Taru Publications

[4] R-forge distributions Core Team Handbook on probability distributions University Year 2009-2010

[5] Johnson, Norman L.; Kotz, Samuel; Balakrishnan, N. (1994), *Continuous univariate distributions. Vol. 1*, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics (2nd ed.), New York: John Wiley & Sons, pp. 284–285, ISBN 978-0-471-58495-7, MR 1299979

[6] Barndorff-Nielsen O., Blsild P., Halgreen C. First hitting time models for the generalized inverse Gaussian distribution Stochastic Process. Appl., 7 (1) (1978), pp. 49-54

[7] Mimi Zhang, Matthew Revie, John Quigley, Saddlepoint approximation for the generalized inverse Gaussian Lévy process, Journal of Computational and Applied Mathematics, Volume 411, 2022,

[8] Jørgensen, B. (2012). Statistical Properties of the Generalized Inverse Gaussian Distribution. Lecture notes in statistics. Springer Science & Business Media.

[9] https://www.google.com/books/edition/Recent_Developments_in_Multivariate_and/dNH9DwAAQBAJ?hl=en&gbpv=1&dq=%22Generalized+integer+gamma+distribution%22+-%22a+basis+for+distributions+in%22&pg=PA37&printsec=frontcover