# Inverse Gaussian distribution / Wald distribution

The Inverse Gaussian Distribution, also known as the Wald distribution or normal-inverse Gaussian, is an exponential distribution that is commonly used to model non-negative, positively skewed data. It is characterized by a slow decrease in its tail compared to the normal (Gaussiam) distribution, making it suitable for modeling phenomena where extremely large values are more likely. The distribution has a single mode and finds wide applications in various fields such as business, survival analysis, finance, medicine, and labor dispute resolution.

Despite its name, the term “inverse” in the “Inverse Gaussian” does not refer to the inverse of a distribution. In addition, the Inverse Gaussian distribution should not be confused with the “inverse normal” procedure for finding z-values — despite the inverse Gaussian also sometimes being called the “normal inverse” distribution. The inverse normal distribution refers to a method of finding probabilities for a normal distribution, whereas the Inverse Gaussian distribution is a probability distribution with a single mode and long tail.

## Inverse Gaussian distribution PDF

The probability density function (PDF) for the inverse Gaussian distribution is:

where λ > 0, μ > 0 and

• μ = location parameter
• λ = scale parameter.

Although the inverse Gaussian distribution has a relatively simple probability density function formula, calculating probabilities numerically still requires careful techniques to achieve high precision for all parameter values when using floating point arithmetic. Multiple R packages provide functions for the inverse Gaussian distribution, including rmutil, SuppDists, invGauss, and LaplacesDemon. These packages implement methods to accurately evaluate the inverse Gaussian distribution for all parameters.

## Comparison to other distributions

Although the Inverse Gaussian distribution has a similar shape to the three-parameter Weibull distribution, it has the advantage of being easier to estimate probabilities.

While the Gamma distribution can resemble the Inverse Gaussian distribution in shape given certain parameters, the Inverse Gaussian distribution is better suited to producing extremely large values. Despite their similar appearances, the Inverse Gaussian distribution has the advantage of more easily generating outliers relative to the Gamma distribution.

## History of the inverse Gaussian distribution

The Inverse Gaussian distribution was first derived independently by multiple mathematicians and scientists in the early 1900s. In 1900, Louis Bachelier [3] derived it as a model for the time a stock reaches a certain price for the first time. In 1915, Erwin Schrödinger [4] and Marian von Smoluchowski [5] derived it as a model for the time to first passage of a Brownian motion process. In reproduction modeling, it is known as the Hadwiger function, described by Hugo Hadwiger in 1940 [6]. Abraham Wald [7] re-derived the distribution in 1944 as the limiting form of a sequential probability ratio test. The name “Inverse Gaussian” was proposed by Maurice Tweedie, who investigated its statistical properties in the 1940s and 1950s [e.g., 1]. The distribution was reviewed in depth by Folks and Chhikara in 1978 [8].

## References

[1] Tweedie, M. C. K. (1956). Some Statistical Properties of Inverse Gaussian Distributions. Virginia Journal of Science, 7, 160-165.

[2] Gerhard Thallinger, CC BY-SA 4.0 https://creativecommons.org/licenses/by-sa/4.0, via Wikimedia Commons

[3] Bachelier, Louis (1900), “The Theory of Speculation”Ann. Sci. Éc. Norm. Supér., Serie 3, 17: 21–89 (Engl. translation by David R. May, 2011), doi:10.24033/asens.476

[4] Schrödinger, Erwin (1915), “Zur Theorie der Fall- und Steigversuche an Teilchen mit Brownscher Bewegung” [On the Theory of Fall- and Rise Experiments on Particles with Brownian Motion], Physikalische Zeitschrift (in German), 16 (16): 289–295

[5] Smoluchowski, Marian (1915), “Notiz über die Berechnung der Brownschen Molekularbewegung bei der Ehrenhaft-Millikanschen Versuchsanordnung” [Note on the Calculation of Brownian Molecular Motion in the Ehrenhaft-Millikan Experimental Set-up], Physikalische Zeitschrift (in German), 16 (17/18): 318–321

[6] Hadwiger, H. (1940). “Eine analytische Reproduktionsfunktion für biologische Gesamtheiten” Skandinavisk Aktuarietidskrijt. 7 (3–4): 101–113. doi:10.1080/03461238.1940.10404802.

[7] Wald, Abraham (1944), “On Cumulative Sums of Random Variables”, Annals of Mathematical Statistics15 (3): 283–296, doi:10.1214/aoms/1177731235JSTOR 2236250

[8] Folks, J. Leroy; Chhikara, Raj S. (1978), “The Inverse Gaussian Distribution and Its Statistical Application—A Review”, Journal of the Royal Statistical Society, Series B (Methodological), 40 (3): 263–275, doi:10.1111/j.2517-6161.1978.tb01039.x, JSTOR 2984691, S2CID 125337421

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