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.

Definition

The F-distribution with ''d''₁ and ''d''₂ degrees of freedom is the distribution of

: $X = \frac$

where $S_1$ and $S_2$ are independent random variables with chi-square distributions with respective degrees of freedom $d_1$ and $d_2$.

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

:
\begin
f(x; d_1,d_2) &= \frac \\ pt&=\frac \left(\frac\right)^ x^ \left(1+\frac \, x \right)^
\end

for real ''x'' > 0. Here $\mathrm$ is the beta function. In many applications, the parameters ''d''₁ and ''d''₂ are positive integers, but the distribution is well-defined for positive real values of these parameters.

The cumulative distribution function is

:$F(x; d_1,d_2)=I_\left (\tfrac, \tfrac \right) ,$

where ''I'' is the regularized incomplete beta function.

The expectation, variance, and other details about the F(''d''₁, ''d''₂) are given in the sidebox; for ''d''₂ > 8, the excess kurtosis is

:$\gamma_2 = 12\frac.$

The ''k''-th moment of an F(''d''₁, ''d''₂) distribution exists and is finite only when 2''k'' < ''d''₂ and it is equal to

:$\mu _X(k) =\left( \frac\right)^k \frac \frac.$  

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

:$\varphi^F_(s) = \frac U \! \left(\frac,1-\frac,-\frac \imath s \right)$

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

Characterization

A random variate of the ''F''-distribution with parameters $d_1$ and $d_2$ arises as the ratio of two appropriately scaled chi-squared variates:

:$X = \frac$

where

*$U_1$ and $U_2$ have chi-squared distributions with $d_1$ and $d_2$ degrees of freedom respectively, and
*$U_1$ and $U_2$ are independent.

In instances where the ''F''-distribution is used, for example in the analysis of variance, independence of $U_1$ and $U_2$ might be demonstrated by applying Cochran's theorem.

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

:$X = \frac \div \frac,$

where $s_1^2 = \frac$ and $s_2^2 = \frac$, $S_1^2$ 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 $N(0,\sigma_2^2)$.

In a frequentist context, a scaled ''F''-distribution therefore gives the probability $p(s_1^2/s_2^2 \mid \sigma_1^2, \sigma_2^2)$, with the ''F''-distribution itself, without any scaling, applying where $\sigma_1^2$ is being taken equal to $\sigma_2^2$. 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 $\sigma_2^2$.G. E. P. Box and G. C. Tiao (1973), ''Bayesian Inference in Statistical Analysis'', Addison-Wesley. p. 110 In this context, a scaled ''F''-distribution thus gives the posterior probability $p(\sigma^2_2 /\sigma_1^2 \mid s^2_1, s^2_2)$, where the observed sums $s^2_1$ and $s^2_2$ are now taken as known.

Properties and related distributions

*If $X \sim \chi^2_$ and $Y \sim \chi^2_$ ( Chi squared distribution) are independent, then $\frac \sim \mathrm(d_1, d_2)$
*If $X_k \sim \Gamma(\alpha_k,\beta_k)\,$ (Gamma distribution) are independent, then $\frac \sim \mathrm(2\alpha_1, 2\alpha_2)$
*If $X \sim \operatorname(d_1/2,d_2/2)$ (Beta distribution) then $\frac \sim \operatorname(d_1,d_2)$
*Equivalently, if $X \sim F(d_1, d_2)$, then $\frac \sim \operatorname(d_1/2,d_2/2)$.
*If $X \sim F(d_1, d_2)$, then $\fracX$ has a beta prime distribution: $\fracX \sim \operatorname\left(\tfrac,\tfrac\right)$.
*If $X \sim F(d_1, d_2)$ then $Y = \lim_ d_1 X$ has the chi-squared distribution $\chi^2_$
*$F(d_1, d_2)$ is equivalent to the scaled Hotelling's T-squared distribution $\frac \operatorname^2 (d_1, d_1 +d_2-1)$.
*If $X \sim F(d_1, d_2)$ then $X^ \sim F(d_2, d_1)$.
*If $X\sim t_$ — Student's t-distribution — then: $\begin X^ &\sim \operatorname(1, n) \\ X^ &\sim \operatorname(n, 1) \end$
*''F''-distribution is a special case of type 6 Pearson distribution
*If $X$ and $Y$ are independent, with $X,Y\sim$ Laplace(''μ'', ''b'') then $\frac \sim \operatorname(2,2)$
*If $X\sim F(n,m)$ then $\tfrac \sim \operatorname(n,m)$ (Fisher's z-distribution)
*The noncentral ''F''-distribution simplifies to the ''F''-distribution if $\lambda=0$.
*The doubly noncentral ''F''-distribution simplifies to the ''F''-distribution if $\lambda_1 = \lambda_2 = 0$
*If $\operatorname_X(p)$ is the quantile ''p'' for $X\sim F(d_1,d_2)$ and $\operatorname_Y(1-p)$ is the quantile $1-p$ for $Y\sim F(d_2,d_1)$, then $\operatorname_X(p)=\frac.$
* ''F''-distribution is an instance of ratio distributions

See also

* Beta prime distribution
* Chi-square distribution
* Chow test
* Gamma distribution
* Hotelling's T-squared distribution
* Wilks' lambda distribution
* Wishart distribution

References

External links

Table of critical values of the ''F''-distributionEarliest Uses of Some of the Words of Mathematics: entry on ''F''-distribution contains a brief history

{{DEFAULTSORT:F-distribution
Continuous distributions
Analysis of variance