Inverse Normal Distribution

Statistics Definitions > Inverse Normal Distribution

What is an Inverse Normal Distribution?

An inverse normal distribution is a way to work backwards from a known probability to find an x-value. It is an informal term and doesn’t refer to a particular probability distribution. There is also a generalized inverse normal  family, proposed by Robert in 1991 [1].

How to Find Inverse Normal on the TI-83 with the InvNorm Command

The InvNorm function on the TI-83 gives you an x-value if you input the area (probability region) to the left of the x-value. The area must be between 0 and 1. You must also input the mean and standard deviation.

Example question: Find the 90th percentile for a normal distribution with a mean of 70 and a standard deviation of 4.5.

  1. Press 2nd then VARS to access the DISTR menu.
  2. Arrow down to 3:invNorm( and press ENTER.
  3. Type the area, mean and standard deviation in the following format:

invNorm (probability,mean,standard deviation).
For this example, your input will look like this:
invNorm(90,70,4,.5).

inverse normal distribution
The x-value (90th percentile) is 75.767.

Robert’s generalized inverse normal distribution

Robert’s generalized inverse normal distribution has a probability density function (pdf) of

Robert’s generalized inverse normal distribution formula

where K(a, ξ, σ) is the normalizing constant, which can be expressed explicitly in terms of confluent hypergeometric functions [2].

This is a bimodal distribution with two modes at

  • x1 = -ξ / (a + 1)
  • x2 = -ξ / (a – 1).

Robert described a second generalized inverse normal 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 pdf of Z = 1/X is

The generalized inverse normal has been used by subsequent authors including Druilhet and Pommeret in their paper on Bayesian analysis [3].

Difference between inverse normal and inverse gaussian

The names “Gaussian Distribution” and “Normal Distribution” mean the same thing (i.e. a bell shaped curve). Physicists use the term Gaussian and Statisticians use the term “Normal.” However, The inverse normal is not the same thing as the Inverse Gaussian distribution.

  • The inverse normal distribution refers to the technique of working backwards to find x-values. In other words, you’re finding the inverse.
  • The inverse Gaussian is a two-parameter family of continuous probability distributions.

The “inverse” in “inverse Gaussian” is misleading because the distribution isn’t actually an inverse. In fact, at large values of it’s shape parameter, the inverse Gaussian looks exactly like the normal distribution.

References

  1. Robert, C. (1991). Generalized inverse normal. Statistics & Probability Letters, 11, 37-41.
  2. Johnson, Kotz, and Balakrishnan, (1994), Continuous Univariate Distributions, Volumes I and II, 2nd. Ed., John Wiley and Sons.
  3. Druilet & Pommeret. (2012). Invariant Conjugate Analysis for Exponential Families. Bayesian Analysis 7, Number 4, pp. 903–916

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