naginterfaces.library.stat.inv_​cdf_​beta

naginterfaces.library.stat.inv_cdf_beta(p, a, b, tol=0.0)

inv_cdf_beta returns the deviate associated with the given lower tail probability of the beta distribution.

For full information please refer to the NAG Library document for g01fe

https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/g01/g01fef.html

Parameters

pfloat

$p$, the lower tail probability from the required beta distribution.

afloat

$a$, the first parameter of the required beta distribution.

bfloat

$b$, the second parameter of the required beta distribution.

tolfloat, optional

The relative accuracy required by you in the result. If inv_cdf_beta 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.

Returns

xfloat

The deviate associated with the given lower tail probability of the beta distribution.

Raises

NagValueError

(errno $1$)

On entry, $p=⟨value⟩$.

Constraint: $p\ge 0.0$.

(errno $1$)

On entry, $p=⟨value⟩$.

Constraint: $p\le 1.0$.

(errno $2$)

On entry, $a=⟨value⟩$ and $b=⟨value⟩$.

Constraint: $a>0.0$.

(errno $2$)

On entry, $a=⟨value⟩$ and $b=⟨value⟩$.

Constraint: $a\le {10}^6$.

(errno $2$)

On entry, $a=⟨value⟩$ and $b=⟨value⟩$.

Constraint: $b>0.0$.

(errno $2$)

On entry, $a=⟨value⟩$ and $b=⟨value⟩$.

Constraint: $b\le {10}^6$.

Warns

NagAlgorithmicWarning

(errno $3$)

The solution has failed to converge. However, the result should be a reasonable approximation. Requested accuracy not achieved when calculating beta probability. You should try setting $tol$ larger.

(errno $4$)

The requested accuracy has not been achieved. Use a larger value of $tol$. There is doubt concerning the accuracy of the computed result. $100$ iterations of the Newton–Raphson method have been performed without satisfying the accuracy criterion (see Further Comments). The result should be a reasonable approximation of the solution.

Notes

The deviate, ${\beta }_p$, associated with the lower tail probability, $p$, of the beta distribution with parameters $a$ and $b$ is defined as the solution to

$$P(B\le {\beta }_p:a,b)=p=\frac{\Gamma (a+b)}{\Gamma \left(a\right)\Gamma \left(b\right)}{\int }_0^{{\beta }_p}{B}^{a-1}{(1-B)}^{b-1}dB\text{,}0\le {\beta }_p\le 1\text{;}a,b>0\text{.}$$

The algorithm is a modified version of the Newton–Raphson method, following closely that of Cran et al. (1977).

An initial approximation, ${\beta }_0$, to ${\beta }_p$ is found (see Cran et al. (1977)), and the Newton–Raphson iteration

$${\beta }_i={\beta }_{i-1}-\frac{f\left({\beta }_{i-1}\right)}{f^{\prime }\left({\beta }_{i-1}\right)}\text{,}$$

where $f\left(\beta \right)=P(B\le \beta :a,b)-p$ is used, with modifications to ensure that $\beta$ remains in the range $(0,1)$.

References

Cran, G W, Martin, K J and Thomas, G E, 1977, Algorithm AS 109. Inverse of the incomplete beta function ratio, Appl. Statist. (26), 111–114

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