Fisher-F-distribution

$F$-distribution, Fisher–Snedecor distribution, Snedecor distribution

A continuous probability distribution concentrated on $(0,\infty)$ with density

\begin{equation} \tag{1} f _ { \nu _ { 1 } , \nu _ { 2 } } ( x ) = \frac { 1 } { B ( \nu _ { 1 } / 2 , \nu _ { 2 } / 2 ) } ( \frac { \nu _ {1 } } { \nu _ { 2 } } ) ^ { \nu _ { 1 } / 2 } \times \end{equation}

\begin{equation*} \times \,x ^ { ( \nu _ { 1 } / 2 ) - 1 } \left( 1 + \frac { \nu _ { 1 } } { \nu _ { 2 } } x \right) ^ { ( \nu _ { 1 } + \nu _ { 2 } ) / 2 } , \quad x > 0, \end{equation*}

where $\nu _ { 1 } , \nu _ { 2 } > 0$ are parameters, and $B ( l _ { 1 } , l _ { 2 } )$ is the beta-function. For $\nu _ { 1 } > 2$ it is a unimodal positive asymmetric distribution with mode at the point $x = [ ( \nu _ { 1 } - 2 ) / \nu _ { 1 } ] . [ \nu _ { 2 } / ( \nu _ { 2 } + 2 ) ]$. Its mathematical expectation and variance are, respectively, equal to

\begin{equation*} \frac { \nu _ { 2 } } { \nu _ { 2 } - 2 } \quad \text { for } \nu _ { 2 } > 2 \end{equation*}

and

\begin{equation*} \frac { 2 \nu_2 ^ { 2 }( \nu _ { 1 } + \nu _ { 2 } - 2 ) } { \nu _ { 1 } ( \nu _ { 2 } - 2 ) ^ { 2 } ( \nu _ { 2 } - 4 ) } \quad \text { for } \nu _ { 2 } > 4. \end{equation*}

The Fisher $F$-distribution reduces to a beta-distribution of the second kind (a type-VI distribution in Pearson's classification). It can be regarded as the distribution of a random variable represented in the form of the quotient

\begin{equation*} F _ { \nu _ { 1 } , \nu _ { 2 } } = \frac { \nu _ { 2 } X _ { 1 }} { \nu _ { 1 } X _ { 2 } } , \end{equation*}

where the independent random variables $X _ { 1 }$ and $X _ { 2 }$ have gamma-distributions (cf. Gamma-distribution) with parameters $\nu _ { 1 } / 2$ and $\nu _ { 2 } / 2$, respectively. The distribution function for $F _ { \nu _ { 1 } , \nu _ { 2 } }$ can be expressed in terms of the distribution function $B _ { l_{1} , l _ { 2 } } ( x )$ of the beta-distribution:

\begin{equation} \tag{2} \mathsf{P} \{ F _ { \nu _ { 1 } , \nu _ { 2 } } < x \} = B _ { \nu _ { 1 } / 2 , \nu _ { 2 } / 2} \left( \frac { ( \nu _ { 1 } / \nu _ { 2 } ) x } { 1 + ( \nu _ { 1 } / \nu _ { 2 } ) x } \right). \end{equation}

This relation is used for calculating the values of the Fisher $F$-distribution by means of tables of the beta-distribution. If $\nu _ { 1 } = m$ and $\nu _ { 2 } = n$ are integers, then the Fisher $F$-distribution with $m$ and $n$ degrees of freedom is the distribution of the $F$-quotient

\begin{equation} \tag{3} F _ { m n } = \frac { \chi _ { m } ^ { 2 } / m } { \chi _ { n } ^ { 2 } / n }, \end{equation}

where $\chi ^ { 2 }_{m}$ and $\chi_n ^ { 2 }$ are independent random variables with "chi-squared" distributions (cf. Chi-squared distribution) with $m$ and $n$ degrees of freedom, respectively.

The Fisher $F$-distribution plays a fundamental role in mathematical statistics and appears in the first place as the distribution of the quotient of two sample variances. Namely, let $X _ { 1 } , \dots , X _ { m }$ and $Y _ { 1 } , \ldots , Y _ { n }$ be samples from normal populations with parameters $( a _ { 1 } , \sigma _ { 1 } ^ { 2 } )$ and $( a _ { 2 } , \sigma _ { 2 } ^ { 2 } )$. The expressions

\begin{equation*} s _ { 1 } ^ { 2 } = \frac { 1 } { m - 1 } \sum _ { i } ( X _ { i } - \overline{X} ) ^ { 2 } \quad \text { and } \quad s _ { 2 } ^ { 2 } = \frac { 1 } { n - 1 } \sum _ { j } ( Y _ { j } - \overline{Y} ) ^ { 2 }, \end{equation*}

where $\overline{X} = \sum _ { i } X _ { i } / m$, $\overline{Y} = \sum _ { j } Y _ { j } / n$, serve as estimators of the variances $\sigma _ { 1 } ^ { 2 }$ and $\sigma _ { 2 } ^ { 2 }$. Then the so-called dispersion proportion $F = \sigma _ { 2 } ^ { 2 } s _ { 1 } ^ { 2 } / \sigma _ { 1 } ^ { 2 } s _ { 2 } ^ { 2 }$ has a Fisher $F$-distribution with $m - 1$ and $n - 1$ degrees of freedom under the hypothesis that $\sigma _ { 1 } = \sigma _ { 2 }$ (in this capacity the Fisher $F$-distribution is also called the distribution of the dispersion proportion). The $F$-test is based on the statistic $F$, and it is used, in particular, for testing the hypothesis that the variances of two populations are equal, in the analysis of variance, regression analysis and multi-dimensional statistical analysis.

The universality of the Fisher $F$-distribution is underlined by its connections with other distributions. For $m = 1$ the square of $F _ { m n }$ in (3) has a Student distribution with $n$ degrees of freedom. There are a number of approximations of the Fisher $F$-distribution using the normal and "chi-squared" distributions.

The introduction of the Fisher $F$-distribution in the analysis of variance is connected with the name of R.A. Fisher (1924), although Fisher himself used a quantity $z$ for the dispersion proportion, connected with $F$ by the relation $z = ( \operatorname { log } F ) / 2$. The distribution of $z$ was tabulated by Fisher, and the Fisher $F$-distribution by G. Snedecor (1937). At present the simpler Fisher $F$-distribution is preferred, making use of its connection with the beta-distribution and tables of the incomplete beta-function.

See also Dispersion analysis; Fisher $z$-distribution.

Comments

The dispersion proportion is also known as the variance ratio, and is in the case of the $F$-distribution also called the $F$-ratio. Cf. also Dispersion proportion.

References

[1]	R.A. Fisher, "On a distribution yielding the error functions of several well-known statistics" , Proc. Internat. Congress mathematicians (Toronto 1924) , 2 , Univ. Toronto Press (1928) pp. 805–813
[2]	M.G. Kendall, A. Stuart, "The advanced theory of statistics. Distribution theory" , 3. Design and analysis , Griffin (1969)
[3]	H. Scheffé, "The analysis of variance" , Wiley (1959) Zbl 0086.34603
[4]	L.N. Bol'shev, N.V. Smirnov, "Tables of mathematical statistics" , Libr. math. tables , 46 , Nauka (1983) (In Russian) (Processed by L.S. Bark and E.S. Kedrova)