naginterfaces.library.stat.inv_​cdf_​f_​vector

naginterfaces.library.stat.inv_cdf_f_vector(tail, p, df1, df2)

inv_cdf_f_vector returns a number of deviates associated with given probabilities of the $F$ or variance-ratio distribution with real degrees of freedom.

For full information please refer to the NAG Library document for g01td

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/g01/g01tdf.html

Parameters

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

Indicates which tail the supplied probabilities represent. For $j=\left(mod(\text{i}-1,\text{ltail})\right)+1-1$, for $\text{i}=1,2,\dots ,max(\text{ltail},\text{lp},\text{ldf1},\text{ldf2})$:

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

The lower tail probability, i.e., $p_i=(F_i\le f_{p_i}:u_i,v_i)$.

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

The upper tail probability, i.e., $p_i=(F_i\ge f_{p_i}:u_i,v_i)$.

pfloat, array-like, shape $\left(\text{lp}\right)$

$p_i$, the probability of the required $F$-distribution as defined by $tail$.

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

ffloat, ndarray, shape $(max(\text{ltail},\text{lp}))$

$f_{p_i}$, the deviates for the $F$-distribution.

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

$ivalid[i-1]$ 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 $f_{p_i}$.

$ivalid[i-1]=2$

On entry, invalid value for $p_i$.

$ivalid[i-1]=3$

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

$ivalid[i-1]=4$

The solution has not converged. The result should still be a reasonable approximation to the solution.

$ivalid[i-1]=5$

The value of $p_i$ is too close to $0.0$ or $1.0$ for the result to be computed. This will only occur when the large sample approximations are used.

Raises

NagValueError

(errno $2$)

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

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

(errno $3$)

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

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

(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 $tail$, $p$, $df1$, $df2$ was invalid, or the solution failed to converge.

Check $ivalid$ for more information.

Notes

The deviate, $f_{p_i}$, associated with the lower tail probability, $p_i$, of the $F$-distribution with degrees of freedom $u_i$ and $v_i$ is defined as the solution to

$$(F_i\le f_{p_i}:u_i,v_i)=p_i=\frac{u_i^{\frac{1}{2}{u}_i}{v}_i^{\frac{1}{2}{v}_i}\Gamma \left(\frac{u_i+v_i}{2}\right)}{\Gamma \left(\frac{u_i}{2}\right)\Gamma \left(\frac{v_i}{2}\right)}{\int }_0^{f_{p_i}}{F}_i^{\frac{1}{2}(u_i-2)}{(v_i+u_i{F}_i)}^{-\frac{1}{2}(u_i+v_i)}d{F}_i\text{,}$$

where $u_i,v_i>0$; $0\le f_{p_i}<\infty$.

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

$$(F_i\le f_{p_i}:u_i,v_i)=Pi{\beta }_i(B_i\le \frac{u_i{f}_{p_i}}{u_i{f}_{p_i}+v_i}:u_i/2,v_i/2)$$

and using a call to inv_cdf_beta_vector().

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