F-distribution

Template:Probability distribution{(d_1 x+d_2)^{d_1+d_2}}}}{x\,\mathrm{B}\!\left(\frac{d_1}{2},\frac{d_2}{2}\right)}\!</math> | cdf = <math>I_{\frac{d_1 x}{d_1 x + d_2}} \left(\tfrac{d_1}{2}, \tfrac{d_2}{2} \right)</math> | mean = <math>\frac{d_2}{d_2-2}\!</math>
for d₂ > 2 | median = | mode = <math>\frac{d_1-2}{d_1}\;\frac{d_2}{d_2+2}</math>
for d₁ > 2 | variance = <math>\frac{2\,d_2^2\,(d_1+d_2-2)}{d_1 (d_2-2)^2 (d_2-4)}\!</math>
for d₂ > 4 | skewness = <math>\frac{(2 d_1 + d_2 - 2) \sqrt{8 (d_2-4)}}{(d_2-6) \sqrt{d_1 (d_1 + d_2 -2)}}\!</math>
for d₂ > 6 | kurtosis = see text | entropy = <math>\begin{align} & \ln \Gamma{\left(\tfrac{d_1}{2} \right)}

+ \ln \Gamma{\left(\tfrac{d_2}{2} \right)}
- \ln \Gamma{\left(\tfrac{d_1+d_2}{2} \right)} \\

&+ \left(1-\tfrac{d_1}{2} \right) \psi{\left(1+\tfrac{d_1}{2} \right)}

- \left(1+\tfrac{d_2}{2} \right) \psi{\left(1+\tfrac{d_2}{2} \right)} \\

&+ \left(\tfrac{d_1 + d_2}{2} \right) \psi{\left(\tfrac{d_1 + d_2}{2} \right)}

+ \ln \frac{d_2}{d_1}

\end{align}</math><ref name=lazo1978entropy>Template:Cite journal</ref> | mgf = does not exist, raw moments defined in text and in <ref name=johnson /><ref name=abramowitz /> | char = see text }}

In probability theory and statistics, the F-distribution or F-ratio, 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 (ANOVA) and other F-tests.<ref name=johnson>Template:Cite book</ref><ref name=abramowitz>Template:Abramowitz Stegun ref</ref><ref>NIST (2006). Engineering Statistics Handbook – F Distribution</ref><ref>Template:Cite book</ref>

Definitions

The F-distribution with d₁ and d₂ degrees of freedom is the distribution of

<math display="block"> X = \frac{U_1/d_1}{U_2/d_2} </math>

where <math display=inline>U_1</math> and <math display=inline>U_2</math> are independent random variables with chi-square distributions with respective degrees of freedom <math display=inline>d_1</math> and <math display=inline>d_2</math>.

It can be shown to follow that the probability density function (pdf) for X is given by

<math display="block">\begin{align} f(x; d_1,d_2) &= \frac{\sqrt{\frac{(d_1x)^{d_1}\,\,d_2^{d_2}} {(d_1x+d_2)^{d_1+d_2}}}} {x\operatorname{B}\left(\frac{d_1}{2},\frac{d_2}{2}\right)} \\[5pt] &=\frac{1}{\operatorname{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{align}</math>

for real x > 0. Here <math>\mathrm{B}</math> 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

<math display="block">F(x; d_1,d_2)=I_{d_1 x/(d_1 x + d_2)}\left (\tfrac{d_1}{2}, \tfrac{d_2}{2} \right) ,</math>

where I is the regularized incomplete beta function.

Properties

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

<math display="block">\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)}.</math>

The k-th moment of an F(d₁, d₂) distribution exists and is finite only when 2k < d₂ and it is equal to<ref name=taboga>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

<math display="block">\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) }.</math>

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.,<ref name=abramowitz />). The correct expression <ref>Phillips, P. C. B. (1982) "The true characteristic function of the F distribution," Biometrika, 69: 261–264 Template:JSTOR</ref> is

<math display="block">\varphi^F_{d_1, d_2}(s) = \frac{\Gamma{\left(\frac{d_1+d_2}{2}\right)}}{\Gamma{\left(\tfrac{d_2}{2}\right)}} U \! \left(\frac{d_1}{2},1-\frac{d_2}{2},-\frac{d_2}{d_1} \imath s \right)</math>

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

Related distributions

Relation to the chi-squared distribution

In instances where the F-distribution is used, for example in the analysis of variance, independence of <math>U_1</math> and <math>U_2</math> (defined above) might be demonstrated by applying Cochran's theorem.

Equivalently, since the chi-squared distribution is the sum of squares of independent standard normal random variables, the random variable of the F-distribution may also be written

<math display="block">X = \frac{s_1^2}{\sigma_1^2} \div \frac{s_2^2}{\sigma_2^2},</math>

where <math>s_1^2 = \frac{S_1^2}{d_1}</math> and <math>s_2^2 = \frac{S_2^2}{d_2}</math>, <math>S_1^2</math> is the sum of squares of <math>d_1</math> random variables from normal distribution <math>N(0,\sigma_1^2)</math> and <math>S_2^2</math> is the sum of squares of <math>d_2</math> random variables from normal distribution <math>N(0,\sigma_2^2)</math>.

In a frequentist context, a scaled F-distribution therefore gives the probability <math>p(s_1^2/s_2^2 \mid \sigma_1^2, \sigma_2^2)</math>, with the F-distribution itself, without any scaling, applying where <math>\sigma_1^2</math> is being taken equal to <math>\sigma_2^2</math>. 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 <math>X</math> has the same distribution in Bayesian statistics, if an uninformative rescaling-invariant Jeffreys prior is taken for the prior probabilities of <math>\sigma_1^2</math> and <math>\sigma_2^2</math>.<ref>Template:Cite book</ref> In this context, a scaled F-distribution thus gives the posterior probability <math>p(\sigma^2_2 /\sigma_1^2 \mid s^2_1, s^2_2)</math>, where the observed sums <math>s^2_1</math> and <math>s^2_2</math> are now taken as known.

In general

• If <math>X \sim \chi^2_{d_1}</math> and <math>Y \sim \chi^2_{d_2}</math> (Chi squared distribution) are independent, then <math> \frac{X / d_1}{Y / d_2} \sim \mathrm{F}(d_1, d_2)</math>
• If <math>X_k \sim \Gamma(\alpha_k,\beta_k)\,</math> (Gamma distribution) are independent, then <math> \frac{\alpha_2\beta_1 X_1}{\alpha_1\beta_2 X_2} \sim \mathrm{F}(2\alpha_1, 2\alpha_2)</math>
• If <math>X \sim \operatorname{Beta}(d_1/2,d_2/2)</math> (Beta distribution) then <math>\frac{d_2 X}{d_1(1-X)} \sim \operatorname{F}(d_1,d_2)</math>
• Equivalently, if <math>X \sim F(d_1, d_2)</math>, then <math>\frac{d_1 X/d_2}{1+d_1 X/d_2} \sim \operatorname{Beta}(d_1/2,d_2/2)</math>.
• If <math>X \sim F(d_1, d_2)</math>, then <math>\frac{d_1}{d_2}X</math> has a beta prime distribution: <math>\frac{d_1}{d_2}X \sim \operatorname{\beta^\prime}\left(\tfrac{d_1}{2},\tfrac{d_2}{2}\right)</math>.
• If <math>X \sim F(d_1, d_2)</math> then <math>Y = \lim_{d_2 \to \infty} d_1 X</math> has the chi-squared distribution <math>\chi^2_{d_1}</math>
• <math>F(d_1, d_2)</math> is equivalent to the scaled Hotelling's T-squared distribution <math>\frac{d_2}{d_1(d_1+d_2-1)} \operatorname{T}^2 (d_1, d_1 +d_2-1) </math>.
• If <math>X \sim F(d_1, d_2)</math> then <math>X^{-1} \sim F(d_2, d_1)</math>.
• If <math>X\sim t_{(n)}</math> — Student's t-distribution — then: <math display="block">\begin{align}

X^{2} &\sim \operatorname{F}(1, n) \\ X^{-2} &\sim \operatorname{F}(n, 1) \end{align}</math>

• F-distribution is a special case of type 6 Pearson distribution
• If <math>X</math> and <math>Y</math> are independent, with <math>X,Y\sim</math> Laplace(μ, b) then <math display="block"> \frac{|X-\mu|}{|Y-\mu|} \sim \operatorname{F}(2,2) </math>
• If <math>X\sim F(n,m)</math> then <math>\tfrac{\log{X}}{2} \sim \operatorname{FisherZ}(n,m)</math> (Fisher's z-distribution)
• The noncentral F-distribution simplifies to the F-distribution if <math>\lambda=0</math>.
• The doubly noncentral F-distribution simplifies to the F-distribution if <math> \lambda_1 = \lambda_2 = 0 </math>
• If <math>\operatorname{Q}_X(p)</math> is the quantile p for <math>X\sim F(d_1,d_2)</math> and <math>\operatorname{Q}_Y(1-p)</math> is the quantile <math>1-p</math> for <math>Y\sim F(d_2,d_1)</math>, then <math display="block">\operatorname{Q}_X(p)=\frac{1}{\operatorname{Q}_Y(1-p)}.</math>
• F-distribution is an instance of ratio distributions
• W-distribution<ref>Template:Cite journal</ref> is a unique parametrization of F-distribution.

See also

• Beta prime distribution
• Chi-square distribution
• Chow test
• Gamma distribution
• Hotelling's T-squared distribution
• Wilks' lambda distribution
• Wishart distribution
• Modified half-normal distribution<ref name="Sun, Kong and Pal">Template:Cite journal</ref> with the pdf on <math>(0, \infty)</math> is given as <math> f(x)= \frac{2\beta^{\frac{\alpha}{2}} x^{\alpha-1} \exp(-\beta x^2+ \gamma x )}{\Psi{\left(\frac{\alpha}{2}, \frac{ \gamma}{\sqrt{\beta}}\right)}}</math>, where <math>\Psi(\alpha,z)={}_1\Psi_1\left(\begin{matrix}\left(\alpha,\frac{1}{2}\right)\\(1,0)\end{matrix};z \right)</math> denotes the Fox–Wright Psi function.

External links

• Table of critical values of the F-distribution
• Earliest Uses of Some of the Words of Mathematics: entry on F-distribution contains a brief history
• Free calculator for F-testing