F-distribution

Fisher–Snedecor
Fisher–Snedecor
	Probability density functionLua error in package.lua at line 80: module 'Module:InfoboxImage/data' not found.
	Cumulative distribution functionLua error in package.lua at line 80: module 'Module:InfoboxImage/data' not found.
Parameters	d1, d2 > 0 deg. of freedom
Support	x ∈ (0, +∞) if d1 = 1, otherwise x ∈ [0, +∞)
PDF	(d1x)d1d2d2(d1x+d2)d1+d2xB(d12,d22)
CDF	Id1xd1x+d2(d12,d22)
Mean	d2d2−2 for d2 > 2
Mode	d1−2d1d2d2+2; for d1 > 2
Variance	2d22(d1+d2−2)d1(d2−2)2(d2−4) for d2 > 4
Skewness	(2d1+d2−2)8(d2−4)(d2−6)d1(d1+d2−2) for d2 > 6
Excess kurtosis	see text
Entropy	ln⁡Γ(d12)+ln⁡Γ(d22)−ln⁡Γ(d1+d22)+(1−d12)ψ(1+d12)−(1+d22)ψ(1+d22)+(d1+d22)ψ(d1+d22)+ln⁡d2d1
MGF	does not exist, raw moments defined in text and in
CF	see text

Template:Short description Script error: No such module "about". Script error: No such module "Distinguish".

Page Module:Infobox/styles.css has no content.Page Template:Infobox probability distribution/styles.css has no content.

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.^[2]^[3]^[4]^[5]

Definitions

The F-distribution with ⁠ $d_{1}$ ⁠ and ⁠ $d_{2}$ ⁠ 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 ⁠ $d_{1}$ ⁠ and ⁠ $d_{2}$ ⁠.

It can be shown to follow that the probability density function (PDF) for ⁠ $X$ ⁠ is given by $\begin{matrix} f (x; d_{1}, d_{2}) & = \frac{\sqrt{\frac{(d_{1} x)^{d_{1}} d_{2}^{d_{2}}}{(d_{1} x + d_{2})^{d_{1} + d_{2}}}}}{x B (\frac{d_{1}}{2}, \frac{d_{2}}{2})} \\ = \frac{{(\frac{d_{1}}{d_{2}})}^{\frac{d_{1}}{2}} x^{\frac{d_{1}}{2} - 1} {(1 + \frac{d_{1}}{d_{2}} x)}^{- \frac{d_{1} + d_{2}}{2}}}{B (\frac{d_{1}}{2}, \frac{d_{2}}{2})} \end{matrix}$

for real ⁠ $x > 0$ ⁠. Here $B$ 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 $F (x; d_{1}, d_{2}) = I_{\frac{d_{1} x}{(d_{1} x + d_{2})}} (\frac{d_{1}}{2}, \frac{d_{2}}{2}),$

where ⁠ $I_{x} (a, b)$ ⁠ is the regularized incomplete beta function.

Properties

The expectation, variance, and other details about the F-distribution ⁠ $F (d_{1}, d_{2})$ ⁠ are given in the sidebox; for ⁠ $d_{2} > 8$ ⁠, the excess kurtosis is $γ_{2} = 12 \frac{d_{1} (5 d_{2} - 22) (d_{1} + d_{2} - 2) + (d_{2} - 4) (d_{2} - 2)^{2}}{d_{1} (d_{2} - 6) (d_{2} - 8) (d_{1} + d_{2} - 2)} .$

The $k$ -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^[6]

$μ_{X} (k) = {(\frac{d_{2}}{d_{1}})}^{k} \frac{Γ (\frac{d_{1}}{2} + k)}{Γ (\frac{d_{1}}{2})} \frac{Γ (\frac{d_{2}}{2} - k)}{Γ (\frac{d_{2}}{2})} .$

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.,^[3]). The correct expression ^[7] is

$φ_{d_{1}, d_{2}}^{F} (s) = \frac{Γ (\frac{d_{1} + d_{2}}{2})}{Γ (\frac{d_{2}}{2})} U (\frac{d_{1}}{2}, 1 - \frac{d_{2}}{2}, - \frac{d_{2}}{d_{1}} ı s)$

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 ⁠ $U_{1}$ ⁠ and ⁠ $U_{2}$ ⁠ (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 $X = \frac{s_{1}^{2}}{σ_{1}^{2}} \div \frac{s_{2}^{2}}{σ_{2}^{2}},$

where $s_{1}^{2} = \frac{S_{1}^{2}}{d_{1}}$ and ⁠ $s_{2}^{2} = \frac{S_{2}^{2}}{d_{2}}$ ⁠, $S_{1}^{2}$ is the sum of squares of $d_{1}$ random variables from normal distribution $N (0, σ_{1}^{2})$ and $S_{2}^{2}$ is the sum of squares of $d_{2}$ random variables from normal distribution ⁠ $N (0, σ_{2}^{2})$ ⁠.

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 the observed sums $s_{1}^{2}$ and $s_{2}^{2}$ are now taken as known.

In general

If $X \sim χ_{d_{1}}^{2}$ and $Y \sim χ_{d_{2}}^{2}$ (Chi squared distribution) are independent, then ⁠ $\frac{X / d_{1}}{Y / d_{2}} \sim F (d_{1}, d_{2})$ ⁠.
If $X_{k} \sim Γ (α_{k}, β_{k})$ (Gamma distribution) are independent, then ⁠ $\frac{α_{2} β_{1} X_{1}}{α_{1} β_{2} X_{2}} \sim F (2 α_{1}, 2 α_{2})$ ⁠.
If X∼Beta⁡(d12,d22) (Beta distribution) then ⁠ $\frac{d_{2} X}{d_{1} (1 - X)} \sim F (d_{1}, d_{2})$ ⁠.
- Equivalently, if ⁠ $X \sim F (d_{1}, d_{2})$ ⁠, then ⁠ $\frac{d_{1} X / d_{2}}{1 + d_{1} X / d_{2}} \sim Beta (\frac{d_{1}}{2}, \frac{d_{2}}{2})$ ⁠.
If ⁠ $X \sim F (d_{1}, d_{2})$ ⁠, then $\frac{d_{1}}{d_{2}} X$ has a beta prime distribution: ⁠ $\frac{d_{1}}{d_{2}} X \sim β^{'} (\frac{d_{1}}{2}, \frac{d_{2}}{2})$ ⁠.
If $X \sim F (d_{1}, d_{2})$ then $Y = lim_{d_{2} \to \infty} d_{1} X$ has the chi-squared distribution ⁠ $χ_{d_{1}}^{2}$ ⁠.
$F (d_{1}, d_{2})$ is equivalent to the scaled Hotelling's T-squared distribution ⁠ $\frac{d_{2}}{d_{1} (d_{1} + d_{2} - 1)} T^{2} (d_{1}, d_{1} + d_{2} - 1)$ ⁠.
If $X \sim F (d_{1}, d_{2})$ then ⁠ $X^{- 1} \sim F (d_{2}, d_{1})$ ⁠.
If ⁠ $X \sim t_{(n)}$ ⁠ – Student's t-distribution – then: $\begin{matrix} X^{2} & \sim F (1, n), \\ X^{- 2} & \sim F (n, 1) . \end{matrix}$
F-distribution is a special case of type 6 Pearson distribution.
If $X$ and $Y$ are independent, with ⁠ $X, Y \sim Laplace (μ, b)$ ⁠ (Laplace distribution), then $\frac{| X - μ |}{| Y - μ |} \sim F (2, 2) .$
If $X \sim F (n, m)$ then $\frac{\log X}{2} \sim FisherZ (n, m)$ (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 ⁠ $λ_{1} = λ_{2} = 0$ ⁠
If $Q_{X} (p)$ is the quantile ⁠ $p$ ⁠ for $X \sim F (d_{1}, d_{2})$ and $Q_{Y} (1 - p)$ is the quantile $1 - p$ for ⁠ $Y \sim F (d_{2}, d_{1})$ ⁠, then $Q_{X} (p) = \frac{1}{Q_{Y} (1 - p)} .$
F-distribution is an instance of ratio distributions.
W-distribution is a unique parametrization of F-distribution.

References

Page Template:Reflist/styles.css has no content.

^ Page Module:Citation/CS1/styles.css has no content.Lazo, A.V.; Rathie, P. (1978). "On the entropy of continuous probability distributions". IEEE Transactions on Information Theory. 24 (1). IEEE: 120–122. doi:10.1109/tit.1978.1055832.
^ ^a ^b Page Module:Citation/CS1/styles.css has no content.Johnson, Norman Lloyd; Samuel Kotz; N. Balakrishnan (1995). Continuous Univariate Distributions, Volume 2 (Section 27) (2nd ed.). Wiley. ISBN 0-471-58494-0.
^ ^a ^b ^c Page Module:Citation/CS1/styles.css has no content.Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 26". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 946. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. Template:LCCN.
^ NIST (2006). Engineering Statistics Handbook – F Distribution
^ Page Module:Citation/CS1/styles.css has no content.Mood, Alexander; Franklin A. Graybill; Duane C. Boes (1974). Introduction to the Theory of Statistics (Third ed.). McGraw-Hill. pp. 246–249. ISBN 0-07-042864-6.
^ Page Module:Citation/CS1/styles.css has no content.Taboga, Marco. "The F distribution".
^ Phillips, P. C. B. (1982) "The true characteristic function of the F distribution," Biometrika, 69: 261–264 JSTOR 2335882
^ Page Module:Citation/CS1/styles.css has no content.Box, G. E. P.; Tiao, G. C. (1973). Bayesian Inference in Statistical Analysis. Addison-Wesley. p. 110. ISBN 0-201-00622-7.

External links

Lua error in package.lua at line 80: module 'Module:Navbox/configuration' not found.

[lazo1978entropy-1] Page Module:Citation/CS1/styles.css has no content.Lazo, A.V.; Rathie, P. (1978). "On the entropy of continuous probability distributions". IEEE Transactions on Information Theory. 24 (1). IEEE: 120–122. doi:10.1109/tit.1978.1055832.

[johnson-2] Page Module:Citation/CS1/styles.css has no content.Johnson, Norman Lloyd; Samuel Kotz; N. Balakrishnan (1995). Continuous Univariate Distributions, Volume 2 (Section 27) (2nd ed.). Wiley. ISBN 0-471-58494-0.

[abramowitz-3] Page Module:Citation/CS1/styles.css has no content.Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 26". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 946. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. Template:LCCN.

[4] NIST (2006). Engineering Statistics Handbook – F Distribution

[5] Page Module:Citation/CS1/styles.css has no content.Mood, Alexander; Franklin A. Graybill; Duane C. Boes (1974). Introduction to the Theory of Statistics (Third ed.). McGraw-Hill. pp. 246–249. ISBN 0-07-042864-6.

[taboga-6] Page Module:Citation/CS1/styles.css has no content.Taboga, Marco. "The F distribution".

[7] Phillips, P. C. B. (1982) "The true characteristic function of the F distribution," Biometrika, 69: 261–264 JSTOR 2335882

[8] Page Module:Citation/CS1/styles.css has no content.Box, G. E. P.; Tiao, G. C. (1973). Bayesian Inference in Statistical Analysis. Addison-Wesley. p. 110. ISBN 0-201-00622-7.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

Fisher–Snedecor
Probability density function Lua error in package.lua at line 80: module 'Module:InfoboxImage/data' not found.
Cumulative distribution function Lua error in package.lua at line 80: module 'Module:InfoboxImage/data' not found.
Parameters	$d 1$ , $d 2 > 0$ deg. of freedom
Support	$x \in (0, +\infty)$ if $d 1 = 1$ , otherwise $x \in [0, +\infty)$
PDF	$\frac{\sqrt{\frac{(d_{1} x)^{d_{1}} d_{2}^{d_{2}}}{(d_{1} x + d_{2})^{d_{1} + d_{2}}}}}{x B (\frac{d_{1}}{2}, \frac{d_{2}}{2})}$
CDF	$I_{\frac{d_{1} x}{d_{1} x + d_{2}}} (\frac{d_{1}}{2}, \frac{d_{2}}{2})$
Mean	$\frac{d_{2}}{d_{2} - 2}$ for $d 2 > 2$
Mode	$\frac{d_{1} - 2}{d_{1}} \frac{d_{2}}{d_{2} + 2}$ for $d 1 > 2$
Variance	$\frac{2 d_{2}^{2} (d_{1} + d_{2} - 2)}{d_{1} (d_{2} - 2)^{2} (d_{2} - 4)}$ for $d 2 > 4$
Skewness	$\frac{(2 d_{1} + d_{2} - 2) \sqrt{8 (d_{2} - 4)}}{(d_{2} - 6) \sqrt{d_{1} (d_{1} + d_{2} - 2)}}$ for $d 2 > 6$
Excess kurtosis	see text
Entropy	$\begin{matrix} \ln Γ (\frac{d_{1}}{2}) + \ln Γ (\frac{d_{2}}{2}) - \ln Γ (\frac{d_{1} + d_{2}}{2}) \\ + (1 - \frac{d_{1}}{2}) ψ (1 + \frac{d_{1}}{2}) \\ - (1 + \frac{d_{2}}{2}) ψ (1 + \frac{d_{2}}{2}) \\ + (\frac{d_{1} + d_{2}}{2}) ψ (\frac{d_{1} + d_{2}}{2}) + \ln \frac{d_{2}}{d_{1}} \end{matrix}$ ^[1]
MGF	does not exist, raw moments defined in text and in ^[2]^[3]
CF	see text

Anonymous

Search

F-distribution

Namespaces

More

Page actions

Contents

Definitions

Properties

Related distributions

Relation to the chi-squared distribution

In general

See also

References

External links

Navigation

Navigation

Contribute

In other projects

Wiki tools

Wiki tools

Anonymous

Search

F-distribution

Definitions

Properties

Related distributions

Relation to the chi-squared distribution

In general

See also

References

External links

Navigation

Wiki tools

Page tools

Other projects

Categories