📂Probability Distribution

Nonsymmetric F-distribution

Definition

Singly Non-central F-distribution ¹

The degrees of freedom $r_{1} , r_{2} > 0$ and non-centrality $\lambda_{1} \ge 0$ define the probability density function of a continuous probability distribution $F \left( r_{1} , r_{2} , \lambda_{1} \right)$, known as the Singly Non-central F-distribution.

$$ f(x) = \sum_{k=0}^{\infty} {{ e^{ - \lambda / 2 } \left( \lambda / 2 \right)^{k} } \over { B \left( {{ r_{2} } \over { 2 }} , {{ r_{1} } \over { 2 }} + k \right) k ! }} \left( {{ r_{1} } \over { r_{2} }} \right)^{{{ r_{1} } \over { 2 }} + k} \left( {{ r_{2} } \over { r_{1} x + r_{2} }} \right) ^{{{ r_{1} + r_{2} } \over { 2 }} + k} x^{{{ r_{1} } \over { 2 }} - 1 + k} \qquad , x \ge 0 $$

Doubly Non-central F-distribution ²

The degrees of freedom $r_{1} , r_{2} > 0$ and non-centrality $\lambda_{1}, \lambda_{2} \ge 0$ define the probability density function of a continuous probability distribution $F \left( r_{1} , r_{2} , \lambda_{1}, \lambda_{2} \right)$, known as the Doubly Non-central F-distribution.

$$ f(x) = \sum_{k=0}^{\infty} \sum_{l=0}^{\infty} {{ n_{1}^{k + {{ r_{1} } \over { 2 }}} n_{2}^{l + {{ r_{2} } \over { 2 }}} x^{k + {{ n_{1} } \over { 2 }} - 1 } \lambda_{1}^{k} \lambda_{2}^{l} } \over { 2^{k+l} k!! e^{{{ \lambda_{1} + \lambda_{2} } \over { 2 }}} B \left( k + {{ 1 } \over { 2 }} r_{1} , l + {{ 1 } \over { 2 }} r_{2} \right) }} \qquad , x \ge 0 $$

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• $B$ is the Beta function.
• $k!!$ is the double factorial of $k$.

Description

The non-central F-distribution is a generalization of the F-distribution, where the singly form involves only the numerator, while the doubly form involves both the numerator and denominator following the non-central chi-squared distribution. The term non-centrality originates from the intuitive derivation of the non-central chi-squared distribution, where the mean of the random variables following the normal distribution is not $0$.

Derivation from the non-central chi-squared distribution

Let $X$ follow the non-central chi-squared distribution $\chi^{2} \left( r_{1} , \lambda_{1} \right)$ and $Y$ follow the chi-squared distribution $\chi^{2} \left( r_{2} \right)$. Then, $$ {{ X / r_{1} } \over { Y / r_{2} }} $$ follows the singly non-central F-distribution. If $Y \sim \chi^{2} \left( r_{2} , \lambda_{2} \right)$, then the random variable follows the doubly non-central F-distribution. In other words, if only the numerator follows the non-central chi-squared distribution, it is singly; if both numerator and denominator do, it is doubly.

See Also

F-distribution

non-central chi-squared distribution