F-distribution

This article is about the central F-distribution. For the generalized distribution, see noncentral F-distribution.

Not to be confused with F-statistics as used in population genetics.

Fisher-Snedecor
Probability density function
Cumulative distribution function
Parameters	d₁, d₂ > 0 deg. of freedom
Support	x ∈ [0, +∞)
PDF	${\frac {{\sqrt {{\frac {(d_{1}\,x)^{{d_{1}}}\,\,d_{2}^{{d_{2}}}}{(d_{1}\,x+d_{2})^{{d_{1}+d_{2}}}}}}}}{x\,{\mathrm {B}}\!\left({\frac {d_{1}}{2}},{\frac {d_{2}}{2}}\right)}}\!$
CDF	$I_{{{\frac {d_{1}x}{d_{1}x+d_{2}}}}}\left({\tfrac {d_{1}}{2}},{\tfrac {d_{2}}{2}}\right)$
Mean	${\frac {d_{2}}{d_{2}-2}}\!$ for d₂ > 2
Mode	${\frac {d_{1}-2}{d_{1}}}\;{\frac {d_{2}}{d_{2}+2}}$ for d₁ > 2
Variance	${\frac {2\,d_{2}^{2}\,(d_{1}+d_{2}-2)}{d_{1}(d_{2}-2)^{2}(d_{2}-4)}}\!$ for d₂ > 4
Skewness	${\frac {(2d_{1}+d_{2}-2){\sqrt {8(d_{2}-4)}}}{(d_{2}-6){\sqrt {d_{1}(d_{1}+d_{2}-2)}}}}\!$ for d₂ > 6
Ex. kurtosis	see text
MGF	does not exist, raw moments defined in text and in ^[1]^[2]
CF	see text

Biologist and statistician Ronald Fisher

In probability theory and statistics, the F-distribution, also known as Snedecor's F distribution or the Fisher–Snedecor distribution (after Ronald Fisher and George W. Snedecor) is a continuous probability distribution that arises frequently as the null distribution of a test statistic, most notably in the analysis of variance, e.g., F-test.^[1]^[2]^[3]^[4]

Definition

If a random variable X has an F-distribution with parameters d₁ and d₂, we write X ~ F(d₁, d₂). Then the probability density function (pdf) for X is given by

{\begin{aligned}f(x;d_{1},d_{2})&={\frac {{\sqrt {{\frac {(d_{1}\,x)^{{d_{1}}}\,\,d_{2}^{{d_{2}}}}{(d_{1}\,x+d_{2})^{{d_{1}+d_{2}}}}}}}}{x\,{\mathrm {B}}\!\left({\frac {d_{1}}{2}},{\frac {d_{2}}{2}}\right)}}\\&={\frac {1}{{\mathrm {B}}\!\left({\frac {d_{1}}{2}},{\frac {d_{2}}{2}}\right)}}\left({\frac {d_{1}}{d_{2}}}\right)^{{{\frac {d_{1}}{2}}}}x^{{{\frac {d_{1}}{2}}-1}}\left(1+{\frac {d_{1}}{d_{2}}}\,x\right)^{{-{\frac {d_{1}+d_{2}}{2}}}}\end{aligned}}

for real x ≥ 0. Here $\mathrm{B}$ is the beta function. In many applications, the parameters d₁ and d₂ are positive integers, but the distribution is well-defined for positive real values of these parameters.

The cumulative distribution function is

F(x;d_{1},d_{2})=I_{{{\frac {d_{1}x}{d_{1}x+d_{2}}}}}\left({\tfrac {d_{1}}{2}},{\tfrac {d_{2}}{2}}\right),

where I is the regularized incomplete beta function.

The expectation, variance, and other details about the F(d₁, d₂) are given in the sidebox; for d₂ > 8, the excess kurtosis is

\gamma _{2}=12{\frac {d_{1}(5d_{2}-22)(d_{1}+d_{2}-2)+(d_{2}-4)(d_{2}-2)^{2}}{d_{1}(d_{2}-6)(d_{2}-8)(d_{1}+d_{2}-2)}}

The k-th moment of an F(d₁, d₂) distribution exists and is finite only when 2k < d₂ and it is equal to ^[5]

\mu _{{X}}(k)=\left({\frac {d_{{2}}}{d_{{1}}}}\right)^{{k}}{\frac {\Gamma \left({\tfrac {d_{1}}{2}}+k\right)}{\Gamma \left({\tfrac {d_{1}}{2}}\right)}}{\frac {\Gamma \left({\tfrac {d_{2}}{2}}-k\right)}{\Gamma \left({\tfrac {d_{2}}{2}}\right)}}

The F-distribution is a particular parametrization of the beta prime distribution, which is also called the beta distribution of the second kind.

The characteristic function is listed incorrectly in many standard references (e.g.,^[2]). The correct expression ^[6] is

\varphi _{{d_{1},d_{2}}}^{F}(s)={\frac {\Gamma ({\frac {d_{1}+d_{2}}{2}})}{\Gamma ({\tfrac {d_{2}}{2}})}}U\!\left({\frac {d_{1}}{2}},1-{\frac {d_{2}}{2}},-{\frac {d_{2}}{d_{1}}}\imath s\right)

where U(a, b, z) is the confluent hypergeometric function of the second kind.

Characterization

A random variate of the F-distribution with parameters d₁ and d₂ arises as the ratio of two appropriately scaled chi-squared variates:^[7]

X={\frac {U_{1}/d_{1}}{U_{2}/d_{2}}}

where

U₁ and U₂ have chi-squared distributions with d₁ and d₂ degrees of freedom respectively, and
U₁ and U₂ are independent.

In instances where the F-distribution is used, for example in the analysis of variance, independence of U₁ and U₂ might be demonstrated by applying Cochran's theorem.

Equivalently, the random variable of the F-distribution may also be written

X={\frac {s_{1}^{2}}{\sigma _{1}^{2}}}\;/\;{\frac {s_{2}^{2}}{\sigma _{2}^{2}}}

where s₁² and s₂² are the sums of squares S₁² and S₂² from two normal processes with variances σ₁² and σ₂² divided by the corresponding number of χ² degrees of freedom, d₁ and d₂ respectively : $s_{1}^{2}={\frac {S_{1}^{2}}{d_{1}}}$ and $s_{2}^{2}={\frac {S_{2}^{2}}{d_{2}}}$ .

In a frequentist context, a scaled F-distribution therefore gives the probability p(s₁²/s₂² | σ₁², σ₂²), with the F-distribution itself, without any scaling, applying where σ₁² is being taken equal to σ₂². This is the context in which the F-distribution most generally appears in F-tests: where the null hypothesis is that two independent normal variances are equal, and the observed sums of some appropriately selected squares are then examined to see whether their ratio is significantly incompatible with this null hypothesis.

The quantity X has the same distribution in Bayesian statistics, if an uninformative rescaling-invariant Jeffreys prior is taken for the prior probabilities of σ₁² and σ₂².^[8] In this context, a scaled F-distribution thus gives the posterior probability p(σ₂²/σ₁²|s₁², s₂²), where now the observed sums s₁² and s₂² are what are taken as known.

Differential equation

The probability density function of the F-distribution is a solution of the following differential equation:

\left\{{\begin{array}{l}2x\left(d_{1}x+d_{2}\right)f'(x)+\left(2d_{1}x+d_{2}d_{1}x-d_{2}d_{1}+2d_{2}\right)f(x)=0,\\[12pt]f(1)={\frac {d_{1}^{{{\frac {d_{1}}{2}}}}d_{2}^{{{\frac {d_{2}}{2}}}}\left(d_{1}+d_{2}\right){}^{{{\frac {1}{2}}\left(-d_{1}-d_{2}\right)}}}{B\left({\frac {d_{1}}{2}},{\frac {d_{2}}{2}}\right)}}\end{array}}\right\}

Properties and related distributions

If $X\sim \chi _{{d_{1}}}^{2}$ and $Y\sim \chi _{{d_{2}}}^{2}$ are independent, then ${\frac {X/d_{1}}{Y/d_{2}}}\sim {\mathrm {F}}(d_{1},d_{2})$
Parametrization 1: If $X_{k}\sim \Gamma (\alpha _{k},\theta _{k})\,$ are independent, then ${\frac {\alpha _{2}\theta _{2}X_{1}}{\alpha _{1}\theta _{1}X_{2}}}\sim \mathrm {F} (2\alpha _{1},2\alpha _{2})$
Parametrization 2: If $X_{k}\sim \Gamma (\alpha _{k},\beta _{k})\,$ are independent, then ${\frac {\alpha _{2}\beta _{1}X_{1}}{\alpha _{1}\beta _{2}X_{2}}}\sim \mathrm {F} (2\alpha _{1},2\alpha _{2})$
If $X\sim \operatorname {Beta}(d_{1}/2,d_{2}/2)$ (Beta distribution) then ${\frac {d_{2}X}{d_{1}(1-X)}}\sim \operatorname {F}(d_{1},d_{2})$
Equivalently, if X ~ F(d₁, d₂), then ${\frac {d_{1}X/d_{2}}{1+d_{1}X/d_{2}}}\sim \operatorname {Beta}(d_{1}/2,d_{2}/2)$ .
If X ~ F(d₁, d₂) then $Y=\lim _{{d_{2}\to \infty }}d_{1}X$ has the chi-squared distribution $\chi _{{d_{1}}}^{2}$
F(d₁, d₂) is equivalent to the scaled Hotelling's T-squared distribution ${\frac {d_{2}}{d_{1}(d_{1}+d_{2}-1)}}\operatorname {T}^{2}(d_{1},d_{1}+d_{2}-1)$ .
If X ~ F(d₁, d₂) then X⁻¹ ~ F(d₂, d₁).
If X ~ t(n) -- Student's t-distribution -- then:

X^{{2}}\sim \operatorname {F}(1,n)

X^{{-2}}\sim \operatorname {F}(n,1)

F-distribution is a special case of type 6 Pearson distribution
If X and Y are independent, with X, Y ~ Laplace(μ, b) then

\tfrac{|X-\mu|}{|Y-\mu|} \sim \operatorname{F}(2,2)

If X ~ F(n, m) then ${\tfrac {\log {X}}{2}}\sim \operatorname {FisherZ}(n,m)$ (Fisher's z-distribution)
The noncentral F-distribution simplifies to the F-distribution if λ = 0.
The doubly noncentral F-distribution simplifies to the F-distribution if $\lambda _{1}=\lambda _{2}=0$
If $\operatorname {Q}_{X}(p)$ is the quantile p for X ~ F(d₁, d₂) and $\operatorname {Q}_{Y}(1-p)$ is the quantile 1−p for Y ~ F(d₂, d₁), then

\operatorname {Q}_{X}(p)={\frac {1}{\operatorname {Q}_{Y}(1-p)}}

References

1 2 Johnson, Norman Lloyd; Samuel Kotz; N. Balakrishnan (1995). Continuous Univariate Distributions, Volume 2 (Second Edition, Section 27). Wiley. ISBN 0-471-58494-0.
1 2 3 Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 26". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C., USA; New York, USA: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 946. ISBN 0-486-61272-4. LCCN 64-60036. MR 0167642. ISBN 978-0-486-61272-0. LCCN 65-12253.
↑ NIST (2006). Engineering Statistics Handbook – F Distribution
↑ Mood, Alexander; Franklin A. Graybill; Duane C. Boes (1974). Introduction to the Theory of Statistics (Third Edition, pp. 246–249). McGraw-Hill. ISBN 0-07-042864-6.
↑ Taboga, Marco. "The F distribution".
↑ Phillips, P. C. B. (1982) "The true characteristic function of the F distribution," Biometrika, 69: 261–264 JSTOR 2335882
↑ M.H. DeGroot (1986), Probability and Statistics (2nd Ed), Addison-Wesley. ISBN 0-201-11366-X, p. 500
↑ G.E.P. Box and G.C. Tiao (1973), Bayesian Inference in Statistical Analysis, Addison-Wesley. p.110

External links

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

Some common univariate probability distributions

Continuous	beta Cauchy chi-squared exponential F gamma Laplace log-normal normal Pareto Student's t uniform Weibull

Discrete	Bernoulli binomial discrete uniform geometric hypergeometric negative binomial Poisson

List of probability distributions

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