naginterfaces.library.stat.prob_​f

naginterfaces.library.stat.prob_f(f, df1, df2, tail='L')

prob_f returns the probability for the lower or upper tail of the $F$ or variance-ratio distribution with real degrees of freedom.

For full information please refer to the NAG Library document for g01ed

https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/g01/g01edf.html

Parameters

ffloat

$f$, the value of the $F$ variate.

df1float

The degrees of freedom of the numerator variance, ${\nu }_1$.

df2float

The degrees of freedom of the denominator variance, ${\nu }_2$.

tailstr, length 1, optional

Indicates whether an upper or lower tail probability is required.

$tail=\text{'L'}$

The lower tail probability is returned, i.e., $P(F\le f:{\nu }_1,{\nu }_2)$.

$tail=\text{'U'}$

The upper tail probability is returned, i.e., $P(F\ge f:{\nu }_1,{\nu }_2)$.

Returns

pfloat

The probability for the lower or upper tail of the $F$ or variance-ratio distribution with real degrees of freedom.

Raises

NagValueError

(errno $1$)

On entry, $tail=⟨value⟩$.

Constraint: $tail=\text{'L'}$ or $\text{'U'}$.

(errno $2$)

On entry, $f=⟨value⟩$.

Constraint: $f\ge 0.0$.

(errno $3$)

On entry, $df1=⟨value⟩$ and $df2=⟨value⟩$.

Constraint: $df1>0.0$ and $df2>0.0$.

(errno $4$)

The probability is too close to $0.0$ or $1.0$. $f$ is too far out into the tails for the probability to be evaluated exactly. The result tends to approach $1.0$ if $f$ is large, or $0.0$ if $f$ is small. The result returned is a good approximation to the required solution.

Notes

The lower tail probability for the $F$, or variance-ratio distribution, with ${\nu }_1$ and ${\nu }_2$ degrees of freedom, $P(F\le f:{\nu }_1,{\nu }_2)$, is defined by:

$$P(F\le f:{\nu }_1,{\nu }_2)=\frac{{\nu }_1^{{\nu }_1/2}{\nu }_2^{{\nu }_2/2}\Gamma \left(({\nu }_1+{\nu }_2)/2\right)}{\Gamma \left({\nu }_1/2\right)\Gamma \left({\nu }_2/2\right)}{\int }_0^f{F}^{({\nu }_1-2)/2}{({\nu }_{1}F+{\nu }_2)}^{-({\nu }_1+{\nu }_2)/2}dF\text{,}$$

for ${\nu }_1$, ${\nu }_2>0$, $f\ge 0$.

The probability is computed by means of a transformation to a beta distribution, $P_{\beta }(B\le \beta :a,b)$:

$$P(F\le f:{\nu }_1,{\nu }_2)=P_{\beta }(B\le \frac{{\nu }_{1}f}{{\nu }_{1}f+{\nu }_2}:{\nu }_1/2,{\nu }_2/2)$$

and using a call to prob_beta().

For very large values of both ${\nu }_1$ and ${\nu }_2$, greater than ${10}^5$, a normal approximation is used. If only one of ${\nu }_1$ or ${\nu }_2$ is greater than ${10}^5$ then a ${\chi }^2$ approximation is used, see Abramowitz and Stegun (1972).

References

Abramowitz, M and Stegun, I A, 1972, Handbook of Mathematical Functions, (3rd Edition), Dover Publications

Hastings, N A J and Peacock, J B, 1975, Statistical Distributions, Butterworth