Fisher distribution

< List of probability distributions

The term Fisher distribution usually refers to one of these probability distributions:

• Fisher’s z-distribution
• F-distribution, also called the Fisher–Snedecor distribution or Fisher F-distribution
• Fisher–Bingham distribution
• Fisher–Tippett (extreme value) distribution
• Fisher’s fiducial distribution

It may also refer to one of these less common distributions:

1. Behrens–Fisher distribution
2. Fisher’s noncentral hypergeometric distribution
3. Von Mises–Fisher distribution on a sphere

Behrens-Fisher distribution

The Behrens-Fisher distribution is a non-central t-distribution that arises in the context of statistical inference when the variances of two normally distributed populations are not assumed to be equal. Specifically, the Behrens-Fisher distribution describes the test statistic that arises when testing the hypothesis that the means of two populations are equal, based on two independent samples of unequal variances.

The probability density function (PDF) of the Behrens-Fisher distribution is given by:

where t is the test statistic, Gamma() is the gamma function, _2F_1() is the hypergeometric function, and m and n are the degrees of freedom for the two samples. A derivation of this PDF can be found in various sources, including Keating [1], Zamani [2] and Lehmann [3].

The Behrens-Fisher distribution is often used in the context of confidence interval estimation and hypothesis testing in the presence of unequal variances. Several methods have been proposed for estimating the parameters of the Behrens-Fisher distribution, including the maximum likelihood method and the method of moments.

References

[1] Keating, J. P. (2018). The Behrens-Fisher problem: A review. Sankhya A, 80(2), 347-380.

[2] Zamani, H., & Amini, M. (2019). Bayesian inference for the Behrens-Fisher problem under various prior specifications. Journal of Statistical Computation and Simulation, 89(7), 1170-1187.

[3] Lehmann, E. L., & Casella, G. (1998). Theory of point estimation (Vol. 2). Springer Science & Business Media.

Fisher’s noncentral hypergeometric distribution

Fisher’s noncentral hypergeometric distribution is a probability distribution that models the number of ‘successes’ in a nonstandard hypergeometric sampling problem. It is named after Sir Ronald A. Fisher, who introduced the distribution in 1941.

The noncentral hypergeometric distribution arises when a sample is drawn without replacement from a finite population of n items containing k ‘successes’ (items possessing a certain attribute of interest) and n – k ‘failures’ (items not possessing that attribute). However, unlike in the classical hypergeometric sampling problem where the sample size n is fixed, in the noncentral hypergeometric distribution, the sample size is itself a random variable with a non-constant probability of selection. Specifically, in Fisher’s formulation, each item has an associated weight, and the probability of selecting an item depends on both its weight and whether it is a ‘success’ or a ‘failure’.

The noncentral hypergeometric distribution has some useful properties. For example, it provides a way to compute the power of a statistical test in the case where the null hypothesis is rejected and the alternative hypothesis involves a shift in the proportion of successes. It also arises in the analysis of certain adaptive sampling designs and in the analysis of certain types of clinical trials.

Mathematically, the probability mass function of Fisher’s noncentral hypergeometric distribution is given by:

for x ∈ [x₁, x_u], where x₁ = max (0, N – (M – (M – n)), x_u = min(N, m),

and with binomial coefficients

Due to its complexity, the noncentral hypergeometric distribution is not as widely used as other probability distributions, such as the binomial and hypergeometric distributions. It is used mostly for tests in contingency tables where a conditional distribution for fixed margins is desired.

Von Mises Fisher Distribution

The Von Mises Fisher Distribution (vMF) is an important isotropic distribution for directional data and statistics. “Directional” statistics are statistics that have direction as well as magnitude, like gene expression data, wind current directions or measurements taken from compasses [1].

The von Mises Fisher Distribution is a generalization of the von Mises distribution to higher dimensions. However, this simple distribution bears many similarities to the multivariate normal distribution. The vMF distribution, like the normal distribution, arises naturally in many situations. For example directional data falls naturally into a vMF distribution, as does data distributed on the unit hypersphere [2].

A unit norm vector x has a von Mises-Fisher distribution when its density is:

P_vmf(x; μ κ) := c_p(κ) e ^κμΤx

Where:

• C^p = the normalizing constant,
• ||μ|| = 1 and κ > 0. [3]
• x ∈ S^p-1 (also written as S_p, this is the p dimensional unit hypersphere).

The normalizing constant (found by integrating polar coordinates) is given by:

Where I_s(Κ) is the modified Bessel function of the first kind.

References

[1] Dhillon, I. & Sra, S. (2003). Modeling Data using Directional Distributions. Retrieved January 15, 2020 from: https://www.cs.utexas.edu/users/inderjit/public_papers/tr03-06.pdf
[2] Banerjee, A. et al. (2005). Clustering on the Unit Hypersphere using von Mises-Fisher Distributions. Journal of Machine Learning Research 6 (2005) 1345–1382.
[3] Sra, S. (2016). Directional Statistics in Machine Learning: a Brief Review. Retrieved January 15, 2019 from: http://arxiv-export-lb.library.cornell.edu/pdf/1605.00316