F-distribution

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.

Definitions

The F-distribution with Template:Tmath and Template:Tmath degrees of freedom is the distribution of $$X=\frac{U_1/d_1}{U_2/d_2}$$

where $U_1$ and $U_2$ are independent random variables with chi-square distributions with respective degrees of freedom Template:Tmath and Template:Tmath.

It can be shown to follow that the probability density function (PDF) for Template:Tmath is given by $$\begin{array}{cc}f(x;d_1,d_2)& =\frac{\sqrt{\frac{(d_{1}x{)}^{d_1}{d}_2^{d_2}}{(d_{1}x+d_2{)}^{d_1+d_2}}}}{x\mathrm{B}(\frac{d_1}{2},\frac{d_2}{2})}\\ & =\frac{{\left(\frac{d_1}{d_2}\right)}^{\frac{d_1}{2}}{x\phantom{\left(\frac{d_1}{d_2}\right)}}^{\frac{d_1}{2}-1}{(1+\frac{d_1}{d_2}x)}^{-\frac{d_1+d_2}{2}}}{\mathrm{B}(\frac{d_1}{2},\frac{d_2}{2})}\end{array}$$

for real Template:Tmath. Here $\mathrm{B}$ is the beta function. In many applications, the parameters Template:Tmath and Template:Tmath are positive integers, but the distribution is well-defined for positive real values of these parameters.

The cumulative distribution function is $$F(x;d_1,d_2)=I_{\frac{d_{1}x}{(d_{1}x+d_2)}}(\frac{d_1}{2},\frac{d_2}{2}),$$

where Template:Tmath is the regularized incomplete beta function.

Properties

The expectation, variance, and other details about the F-distribution Template:Tmath are given in the sidebox; for Template:Tmath, 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 Template:Mvar-th moment of an Template:Tmath distribution exists and is finite only when Template:Tmath and it is equal to

$${\mu }_X(k)={\left(\frac{d_2}{d_1}\right)}^k\frac{\mathrm{\Gamma }(\frac{d_1}{2}+k)}{\mathrm{\Gamma }\left(\frac{d_1}{2}\right)}\frac{\mathrm{\Gamma }(\frac{d_2}{2}-k)}{\mathrm{\Gamma }\left(\frac{d_2}{2}\right)}.$$

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.,). The correct expression is

$${\phi }_{d_1,d_2}^F(s)=\frac{\mathrm{\Gamma }\left(\frac{d_1+d_2}{2}\right)}{\mathrm{\Gamma }\left(\frac{d_2}{2}\right)}U(\frac{d_1}{2},1-\frac{d_2}{2},-\frac{d_2}{d_1}ıs)$$

where Template:Tmath 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 Template:Tmath and Template:Tmath (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 $$X=\frac{s_1^2}{{\sigma }_1^2}÷\frac{s_2^2}{{\sigma }_2^2},$$

where $s_1^2=\frac{S_1^2}{d_1}$ and Template:Tmath, $S_1^2$ is the sum of squares of $d_1$ random variables from normal distribution $N(0,{\sigma }_1^2)$ and $S_2^2$ is the sum of squares of $d_2$ random variables from normal distribution Template:Tmath.

In a frequentist context, a scaled F-distribution therefore gives the probability Template:Tmath, with the F-distribution itself, without any scaling, applying where ${\sigma }_1^2$ is being taken equal to Template:Tmath. 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 ${\sigma }_1^2$ and Template:Tmath. In this context, a scaled F-distribution thus gives the posterior probability Template:Tmath, where the observed sums $s_1^2$ and $s_2^2$ are now taken as known.

In general

• If $X\sim {\chi }_{d_1}^2$ and $Y\sim {\chi }_{d_2}^2$ (Chi squared distribution) are independent, then Template:Tmath.
• If $X_k\sim \mathrm{\Gamma }({\alpha }_k,{\beta }_k)$ (Gamma distribution) are independent, then Template:Tmath.
• If $X\sim \mathrm{Beta}(\frac{d_1}{2},\frac{d_2}{2})$ (Beta distribution) then Template:Tmath.
  • Equivalently, if Template:Tmath, then Template:Tmath.
• If Template:Tmath, then $\frac{d_1}{d_2}X$ has a beta prime distribution: Template:Tmath.
• If $X\sim F(d_1,d_2)$ then $Y=\underset{d_2\to \mathrm{\infty }}{lim}{d}_{1}X$ has the chi-squared distribution Template:Tmath.
• $F(d_1,d_2)$ is equivalent to the scaled Hotelling's T-squared distribution Template:Tmath.
• If $X\sim F(d_1,d_2)$ then Template:Tmath.
• If Template:TmathTemplate:Snd Student's t-distributionTemplate:Snd then: $$\begin{array}{cc}{X}^2& \sim \mathrm{F}(1,n),\\ X^{-2}& \sim \mathrm{F}(n,1).\end{array}$$
• F-distribution is a special case of type 6 Pearson distribution.
• If $X$ and $Y$ are independent, with Template:Tmath (Laplace distribution), then $$\frac{|X-\mu |}{|Y-\mu |}\sim \mathrm{F}(2,2).$$
• If $X\sim F(n,m)$ then $\frac{\log X}{2}\sim \mathrm{FisherZ}(n,m)$ (Fisher's z-distribution).
• The noncentral F-distribution simplifies to the F-distribution if Template:Tmath.
• The doubly noncentral F-distribution simplifies to the F-distribution if Template:Tmath
• If ${\mathrm{Q}}_X(p)$ is the quantile Template:Tmath for $X\sim F(d_1,d_2)$ and ${\mathrm{Q}}_Y(1-p)$ is the quantile $1-p$ for Template:Tmath, then $${\mathrm{Q}}_X(p)=\frac{1}{{\mathrm{Q}}_Y(1-p)}.$$
• F-distribution is an instance of ratio distributions.
• W-distribution 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

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