naginterfaces.library.stat.prob_​f_​vector

naginterfaces.library.stat.prob_f_vector(tail, f, df1, df2)

prob_f_vector returns a number of lower or upper tail probabilities for the $F$ or variance-ratio distribution with real degrees of freedom.

For full information please refer to the NAG Library document for g01sd

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g01/g01sdf.html

Parameters

tailstr, length 1, array-like, shape $\left(\text{ltail}\right)$

Indicates whether the lower or upper tail probabilities are required. For $j=\left(mod(\text{i}-1,\text{ltail})\right)+1-1$, for $\text{i}=1,2,\dots ,max(\text{ltail},\text{lf},\text{ldf1},\text{ldf2})$:

$tail\left[j\right]=\text{'L'}$

The lower tail probability is returned, i.e., $p_i=(F_i\le f_i:u_i,v_i)$.

$tail\left[j\right]=\text{'U'}$

The upper tail probability is returned, i.e., $p_i=(F_i\ge f_i:u_i,v_i)$.

ffloat, array-like, shape $\left(\text{lf}\right)$

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

df1float, array-like, shape $\left(\text{ldf1}\right)$

$u_i$, the degrees of freedom of the numerator variance.

df2float, array-like, shape $\left(\text{ldf2}\right)$

$v_i$, the degrees of freedom of the denominator variance.

Returns

pfloat, ndarray, shape $(max(\text{ltail},\text{lf}))$

$p_i$, the probabilities for the $F$-distribution.

ivalidint, ndarray, shape $(max(\text{ltail},\text{lf}))$

$ivalid[i-1]$ indicates any errors with the input arguments, with

$ivalid[i-1]=0$

No error.

$ivalid[i-1]=1$

On entry, invalid value supplied in $tail$ when calculating $p_i$.

$ivalid[i-1]=2$

On entry, $f_i<0.0$.

$ivalid[i-1]=3$

On entry, $u_i\le 0.0$, or, $v_i\le 0.0$.

$ivalid[i-1]=4$

The solution has failed to converge. The result returned should represent an approximation to the solution.

Raises

NagValueError

(errno $2$)

On entry, $\text{array size}=⟨value⟩$.

Constraint: $\text{ltail}>0$.

(errno $3$)

On entry, $\text{array size}=⟨value⟩$.

Constraint: $\text{lf}>0$.

(errno $4$)

On entry, $\text{array size}=⟨value⟩$.

Constraint: $\text{ldf1}>0$.

(errno $5$)

On entry, $\text{array size}=⟨value⟩$.

Constraint: $\text{ldf2}>0$.

Warns

NagAlgorithmicWarning

(errno $1$)

On entry, at least one value of $f$, $df1$, $df2$ or $tail$ was invalid, or the solution failed to converge.

Check $ivalid$ for more information.

Notes

The lower tail probability for the $F$, or variance-ratio, distribution with $u_i$ and $v_i$ degrees of freedom, $(F_i\le f_i:u_i,v_i)$, is defined by:

$$(F_i\le f_i:u_i,v_i)=\frac{u_i^{u_i/2}{v}_i^{v_i/2}\Gamma \left((u_i+v_i)/2\right)}{\Gamma \left(u_i/2\right)\Gamma \left(v_i/2\right)}{\int }_0^{f_i}{F}_i^{(u_i-2)/2}{(u_i{F}_i+v_i)}^{-(u_i+v_i)/2}d{F}_i\text{,}$$

for $u_i$, $v_i>0$, $f_i\ge 0$.

The probability is computed by means of a transformation to a beta distribution, $P{\beta }_i(B_i\le {\beta }_i:a_i,b_i)$:

$$(F_i\le f_i:u_i,v_i)=P{\beta }_i(B_i\le \frac{u_i{f}_i}{u_i{f}_i+v_i}:u_i/2,v_i/2)$$

and using a call to prob_beta().

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

The input arrays to this function are designed to allow maximum flexibility in the supply of vector arguments by re-using elements of any arrays that are shorter than the total number of evaluations required. See the G01 Introduction for further information.

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