F-distribution

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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

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

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 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.(see discussion)

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.

Generalization

A generalization of the (central) F-distribution is the noncentral F-distribution.

Related distributions and properties

If $X\sim \chi _{{d_{1}}}^{2}$ and $Y\sim \chi _{{d_{2}}}^{2}$ are independent, then ${\frac {X/d_{1}}{Y/d_{2}}}\sim {\mathrm {F}}(d_{1},d_{2})$
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) 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

↑ 1.0 1.1 Johnson, Norman Lloyd; Samuel Kotz, N. Balakrishnan (1995). Continuous Univariate Distributions, Volume 2 (Second Edition, Section 27). Wiley. ISBN 0-471-58494-0. Cite uses deprecated parameters (help)
↑ 2.0 2.1 2.2 Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 26", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, p. 946, ISBN 978-0486612720, MR 0167642 .
↑ NIST (2006). Engineering Statistics Handbook - F Distribution
↑ Mood, Alexander; Franklin A. Graybill, Duane C. Boes (1974). Introduction to the Theory of Statistics (Third Edition, p. 246-249). McGraw-Hill. ISBN 0-07-042864-6. Cite uses deprecated parameters (help)
↑ 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

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, e.g. [0,1]

Arcsine ARGUS Balding–Nichols Bates Beta Beta rectangular Irwin–Hall Kumaraswamy logit-normal Noncentral beta raised cosine Triangular U-quadratic uniform Wigner semicircle Xenakis

[[List of probability distributions#Supported_on_semi-infinite_intervals.2C_usually_.5B0.2C.E2.88.9E.29|Continuous univariate supported on a semi-infinite interval, usually [0,∞)]]

Benini
Benktander 1st kind
Benktander 2nd kind
Beta prime
Burr
chi-squared
chi
Coxian
Dagum
Davis
EL
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
Maxwell–Boltzmann
Maxwell–Jüttner
Mittag–Leffler
Nakagami
noncentral chi-squared
Pareto
phase-type
Poly-Weibull
Rayleigh
relativistic Breit–Wigner
Rice
Rosin–Rammler
shifted Gompertz
truncated normal
type-2 Gumbel
Weibull
Wilks' lambda

Continuous univariate supported on the whole real line (−∞, ∞)

Cauchy exponential power Fisher's z generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson SU Landau Laplace Linnik logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel variance-gamma Voigt

Continuous univariate with support whose type varies

generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential shifted log-logistic

Mixed continuous-discrete univariate distributions

rectified Gaussian

Multivariate (joint)

Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet Generalized Dirichlet multivariate normal Multivariate stable multivariate Student normal-scaled 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 Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular

Degenerate discrete degenerate Dirac delta function Singular Cantor

Families

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

v t e 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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