naginterfaces.library.stat.prob_​f_​noncentral

naginterfaces.library.stat.prob_f_noncentral(f, df1, df2, rlamda, tol=0.0, maxit=500)

prob_f_noncentral returns the probability associated with the lower tail of the noncentral $F$ or variance-ratio distribution.

For full information please refer to the NAG Library document for g01gd

https://support.nag.com/numeric/nl/nagdoc_30.1/flhtml/g01/g01gdf.html

Parameters

ffloat

$f$, the deviate from the noncentral $F$-distribution.

df1float

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

df2float

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

rlamdafloat

$\lambda$, the noncentrality parameter.

tolfloat, optional

The relative accuracy required by you in the results. If prob_f_noncentral is entered with $tol$ greater than or equal to $1.0$ or less than $10\times \text{machine precision}$ (see machine.precision), the value of $10\times \text{machine precision}$ is used instead.

maxitint, optional

The maximum number of iterations to be used.

Returns

pfloat

The probability associated with the lower tail of the noncentral $F$ or variance-ratio distribution.

Raises

NagValueError

(errno $1$)

On entry, $df2=⟨value⟩$.

Constraint: $df2>0.0$.

(errno $1$)

On entry, $rlamda=⟨value⟩$.

Constraint: $0.0\le rlamda\le -2.0\times \log \left(U\right)$, where $U$ is the safe range parameter as defined by machine.real_safe.

(errno $1$)

On entry, $maxit=⟨value⟩$.

Constraint: $maxit\ge 1$.

(errno $1$)

On entry, $df1=⟨value⟩$.

Constraint: $0.0<df1\le {10}^6$.

(errno $1$)

On entry, $f=⟨value⟩$.

Constraint: $f>0.0$.

(errno $1$)

On entry, $df1=⟨value⟩$.

Constraint: $df1>0.0$.

(errno $2$)

The solution has failed to converge in $⟨value⟩$ iterations. Consider increasing $maxit$ or $tol$.

(errno $3$)

The required probability cannot be computed accurately. This may happen if the result would be very close to zero or one. Alternatively the values of $df1$ and $f$ may be too large. In the latter case you could try using a normal approximation, see Abramowitz and Stegun (1972).

Warns

NagAlgorithmicWarning

(errno $4$)

The required accuracy was not achieved when calculating the initial value of the central $F$ or ${\chi }^2$ probability. You should try a larger value of $tol$. If the ${\chi }^2$ approximation is being used then prob_f_noncentral returns zero otherwise the value returned should be an approximation to the correct value.

Notes

The lower tail probability of the noncentral $F$-distribution with ${\nu }_1$ and ${\nu }_2$ degrees of freedom and noncentrality parameter $\lambda$, $P(F\le f:{\nu }_1,{\nu }_2\text{;}\lambda )$, is defined by

$$P(F\le f:{\nu }_1,{\nu }_2\text{;}\lambda )={\int }_0^{x}p(F:{\nu }_1,{\nu }_2\text{;}\lambda )dF\text{,}$$

where

$$P(F:{\nu }_1,{\nu }_2\text{;}\lambda )=\sum _{j=0}^{\infty }{e}^{-\lambda /2}\frac{{\left(\lambda /2\right)}^j}{j!}\times \frac{{({\nu }_1+2j)}^{({\nu }_1+2j)/2}{\nu }_2^{{\nu }_2/2}}{B(({\nu }_1+2j)/2,{\nu }_2/2)}$$

$$\times u^{({\nu }_1+2j-2)/2}{[{\nu }_2+({\nu }_1+2j)u]}^{-({\nu }_1+2j+{\nu }_2)/2}$$

and $B(\cdot ,\cdot )$ is the beta function.

The probability is computed by means of a transformation to a noncentral beta distribution:

$$P(F\le f:{\nu }_1,{\nu }_2\text{;}\lambda )=P_{\beta }(X\le x:a,b\text{;}\lambda )\text{,}$$

where $x=\frac{{\nu }_{1}f}{{\nu }_{1}f+{\nu }_2}$ and $P_{\beta }(X\le x:a,b\text{;}\lambda )$ is the lower tail probability integral of the noncentral beta distribution with parameters $a$, $b$, and $\lambda$.

If ${\nu }_2$ is very large, greater than ${10}^6$, then a ${\chi }^2$ approximation is used.

References

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