F-distribution
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- Not to be confused with F-statistics as used in population genetics.
Probability density function 325px | |
Cumulative distribution function 325px | |
Parameters | d_{1}, d_{2} > 0 deg. of freedom |
Support | x ∈ [0, +∞) |
cdf | |
Mean | for d_{2} > 2 |
Median | |
Mode | for d_{1} > 2 |
Variance | for d_{2} > 4 |
Skewness | for d_{2} > 6 |
Kurtosis | see text |
Entropy | |
mgf | does not exist, raw moments defined in text and in ^{[1]}^{[2]} |
Char. func. | 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.
DefinitionEdit
If a random variable X has an F-distribution with parameters d_{1} and d_{2}, we write X ~ F(d_{1}, d_{2}). Then the probability density function for X is given by
for real x ≥ 0. Here 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
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
- .
The k-th moment of an F(d_{1}, d_{2}) distribution exists and is finite only when 2k < d_{2} and it is equal to ^{[5]}
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
where U(a, b, z) is the confluent hypergeometric function of the second kind.
CharacterizationEdit
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:^{[7]}
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
where s_{1}^{2} and s_{2}^{2} are the sums of squares S_{1}^{2} and S_{2}^{2} from two normal processes with variances σ_{1}^{2} and σ_{2}^{2} divided by the corresponding number of χ^{2} degrees of freedom, d_{1} and d_{2} respectively.
In a Frequentist context, a scaled F-distribution therefore gives the probability p(s_{1}^{2}/s_{2}^{2} | σ_{1}^{2}, σ_{2}^{2}), with the F distribution itself, without any scaling, applying where σ_{1}^{2} is being taken equal to σ_{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 σ_{1}^{2} and σ_{2}^{2}.^{[8]} In this context, a scaled F-distribution thus gives the posterior probability p(σ_{2}^{2}/σ_{1}^{2}|s_{1}^{2}, s_{2}^{2}), where now the observed sums s_{1}^{2} and s_{2}^{2} are what are taken as known.
Generalization Edit
A generalization of the (central) F-distribution is the noncentral F-distribution.
Related distributions and properties Edit
- If and are independent, then
- If (Beta distribution) then
- Equivalently, if X ~ F(d_{1}, d_{2}), then .
- If X ~ F(d_{1}, d_{2}) then has the chi-squared distribution
- F(d_{1}, d_{2}) is equivalent to the scaled Hotelling's T-squared distribution .
- If X ~ F(d_{1}, d_{2}) then X^{−1} ~ F(d_{2}, d_{1}).
- If X ~ t(n) then
- F-distribution is a special case of type 6 Pearson distribution
- If X and Y are independent, with X, Y ~ Laplace(μ, b) then
- If X ~ F(n, m) then (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
- If is the quantile p for X ~ F(d_{1}, d_{2}) and is the quantile 1−p for Y ~ F(d_{2}, d_{1}), then
- .
See also Edit
- Chi-squared distribution
- Chow test
- Gamma distribution
- Hotelling's T-squared distribution
- Student's t-distribution
- Wilks' lambda distribution
- Wishart distribution
References Edit
- ↑ ^{1.0} ^{1.1} Johnson, Norman Lloyd; Samuel Kotz, N. Balakrishnan (1995). Continuous Univariate Distributions, Volume 2 (Second Edition, Section 27), Wiley.
- ↑ ^{2.0} ^{2.1} ^{2.2} Template:Abramowitz Stegun ref
- ↑ 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.
- ↑ The F distribution.
- ↑ Phillips, P. C. B. (1982) "The true characteristic function of the F distribution," Biometrika, 69: 261-264 Template:Jstor
- ↑ 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
Further readingEdit
Key textsEdit
BooksEdit
PapersEdit
Additional materialEdit
BooksEdit
PapersEdit
External linksEdit
- Table of critical values of the F-distribution
- Online significance testing with the F-distribution
- Distribution Calculator Calculates probabilities and critical values for normal, t-, chi2- and F-distribution
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