naginterfaces.library.stat.inv_​cdf_​f

naginterfaces.library.stat.inv_cdf_f(p, df1, df2)

inv_cdf_f returns the deviate associated with the given lower tail probability of the $F$ or variance-ratio distribution with real degrees of freedom.

For full information please refer to the NAG Library document for g01fd

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/g01/g01fdf.html

Parameters

pfloat

$p$, the lower tail probability from the required $F$-distribution.

df1float

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

df2float

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

Returns

xfloat

The deviate associated with the given lower tail probability of the $F$ or variance-ratio distribution with real degrees of freedom.

Raises

NagValueError

(errno $1$)

On entry, $p=⟨value⟩$.

Constraint: $p<1.0$.

(errno $1$)

On entry, $p=⟨value⟩$.

Constraint: $p\ge 0.0$.

(errno $2$)

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 $1.0$. The value of $f_p$ cannot be computed. This will only occur when the large sample approximations are used.

Warns

NagAlgorithmicWarning

(errno $3$)

The solution has failed to converge. However, the result should be a reasonable approximation. Alternatively, inv_cdf_beta() can be used with a suitable setting of the argument $\text{tol}$.

Notes

The deviate, $f_p$, associated with the lower tail probability, $p$, of the $F$-distribution with degrees of freedom ${\nu }_1$ and ${\nu }_2$ is defined as the solution to

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

where ${\nu }_1,{\nu }_2>0$; $0\le f_p<\infty$.

The value of $f_p$ 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 inv_cdf_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