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.NIST (2006). Engineering Statistics Handbook – F Distribution

Definitions The F-distribution with and degrees of freedom is the distribution of

where and are independent random variables with chi-square distributions with respective degrees of freedom and .

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

for real . Here is the beta function. In many applications, the parameters and are positive integers, but the distribution is well-defined for positive real values of these parameters.

The cumulative distribution function is

where is the regularized incomplete beta function.

Properties The expectation, variance, and other details about the F-distribution are given in the sidebox; for , the excess kurtosis is

The -th moment of an distribution exists and is finite only when and it is equal to

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 Phillips, P. C. B. (1982) "The true characteristic function of the F distribution," Biometrika, 69: 261–264 is

where 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 and (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

where and , is the sum of squares of random variables from normal distribution and is the sum of squares of random variables from normal distribution .

In a frequentist context, a scaled F-distribution therefore gives the probability , with the F-distribution itself, without any scaling, applying where is being taken equal to . 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 has the same distribution in Bayesian statistics, if an uninformative rescaling-invariant Jeffreys prior is taken for the prior probabilities of and . In this context, a scaled F-distribution thus gives the posterior probability , where the observed sums and are now taken as known.

In general If and (Chi squared distribution) are independent, then . If (Gamma distribution) are independent, then . If (Beta distribution) then . Equivalently, if , then . If , then has a beta prime distribution: . If then has the chi-squared distribution . is equivalent to the scaled Hotelling's T-squared distribution . If then . If Student's t-distribution then: F-distribution is a special case of type 6 Pearson distribution. If and are independent, with (Laplace distribution), then If then (Fisher's z-distribution). The noncentral F-distribution simplifies to the F-distribution if . The doubly noncentral F-distribution simplifies to the F-distribution if If is the quantile for and is the quantile for , then F-distribution is an instance of ratio distributions. W-distribution is a unique parametrization of F-distribution.

F-distribution

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