Generalized Inverse Normal Distribution


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The term “inverse normal distribution” is informal and doesn’t refer to a specific probability distribution [1]. However, there is an inverse normal density function in statistics that reverses the procedure for finding z-values (see “Inverse Normal Distribution” below). There is a generalized inverse normal distribution family, proposed by Robert in 1991 [2].

Robert’s generalized inverse normal distribution

Robert’s generalized inverse normal distribution has density

K(a, ξ, σ) is the normalizing constant, which can be expressed explicitly in terms of confluent hypergeometric functions [3].

This is a bimodal distribution with two modes at

Robert described a second generalized inverse normal distribution as a generalization of the distribution of 1/X, when X is distributed as a normally distributed random variable with mean 0 and variance σ2. The probability density function(PDF) of Z = 1/X is

The generalized inverse distribution has been used by subsequent authors including Druilhet and Pommeret in their paper on Bayesian analysis [4].

Inverse Normal Distribution

The “inverse normal distribution” refers to the use of an inverse normal density function in statistics to reverse the procedure for finding z-values. Instead of taking a z-score and looking up the percentage area in a table, you take the percentage and look up the z-score. In this sense, the inverse normal distribution is not equal to the Inverse Gaussian distribution, which is a two-parameter family of continuous probability distributions. Perhaps confusingly, the “inverse” in “inverse Gaussian” is misleading because the distribution isn’t actually an inverse. At large values of its shape parameter, the inverse Gaussian approximates the normal distribution.

References

[1] Stephanie Glen. “Inverse Normal Distribution” From StatisticsHowTo.com: Elementary Statistics for the rest of us! https://www.statisticshowto.com/inverse-normal-distribution/

[2] Robert, C. (1991). Generalized inverse normal distributions, Statistics & Probability Letters, 11, 37-41.

[3] Johnson, Kotz, and Balakrishnan, (1994), Continuous Univariate Distributions, Volumes I and II, 2nd. Ed., John Wiley and Sons.

[4] Druilet & Pommeret. (2012). Invariant Conjugate Analysis for Exponential Families. Bayesian Analysis 7, Number 4, pp. 903–916