naginterfaces.library.stat.inv_​cdf_​gamma

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

inv_cdf_gamma returns the deviate associated with the given lower tail probability of the gamma distribution.

For full information please refer to the NAG Library document for g01ff

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g01/g01fff.html

Parameters

pfloat

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

afloat

$\alpha$, the shape parameter of the gamma distribution.

bfloat

$\beta$, the scale parameter of the gamma distribution.

tolfloat, optional

The relative accuracy required by you in the results. The smallest recommended value is $50\times \delta$, where $\delta =max({10}^{-18},\text{machine precision})$. If inv_cdf_gamma is entered with $tol$ less than $50\times \delta$ or greater or equal to $1.0$, then $50\times \delta$ is used instead.

Returns

xfloat

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

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, $a=⟨value⟩$.

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

(errno $2$)

On entry, $b=⟨value⟩$.

Constraint: $b>0.0$.

(errno $2$)

On entry, $a=⟨value⟩$.

Constraint: $a>0.0$.

(errno $3$)

The probability is too close to $0.0$ for the given $a$ to enable the result to be calculated.

(errno $5$)

The series used to calculate the gamma function has failed to converge. This is an unlikely error exit.

Warns

NagAlgorithmicWarning

(errno $4$)

The algorithm has failed to converge in $100$ iterations. A larger value of $tol$ should be tried. The result may be a reasonable approximation.

Notes

The deviate, $g_p$, associated with the lower tail probability, $p$, of the gamma distribution with shape parameter $\alpha$ and scale parameter $\beta$, is defined as the solution to

$$P(G\le g_p:\alpha ,\beta )=p=\frac{1}{{\beta }^{\alpha }\Gamma \left(\alpha \right)}{\int }_0^{g_p}{e}^{-G/\beta }{G}^{\alpha -1}dG\text{,}0\le g_p<\infty \text{;}\alpha ,\beta >0\text{.}$$

The method used is described by Best and Roberts (1975) making use of the relationship between the gamma distribution and the ${\chi }^2$-distribution.

Let $y=2\frac{g_p}{\beta }$. The required $y$ is found from the Taylor series expansion

$$y=y_0+\sum _r\frac{C_r\left(y_0\right)}{r!}{\left(\frac{E}{\varphi \left(y_0\right)}\right)}^r\text{,}$$

where $y_0$ is a starting approximation

$C_1\left(u\right)=1$,

$C_{r+1}\left(u\right)=(r\Psi +\frac{d}{du})C_r\left(u\right)$,

$\Psi =\frac{1}{2}-\frac{\alpha -1}{u}$,

$E=p-{\int }_0^{y_0}\varphi \left(u\right)du$,

$\varphi \left(u\right)=\frac{1}{2^{\alpha }\Gamma \left(\alpha \right)}{e}^{-u/2}{u}^{\alpha -1}$.

For most values of $p$ and $\alpha$ the starting value

$$y_{01}=2\alpha {(z\sqrt{\frac{1}{9\alpha }}+1-\frac{1}{9\alpha })}^3$$

is used, where $z$ is the deviate associated with a lower tail probability of $p$ for the standard Normal distribution.

For $p$ close to zero,

$$y_{02}={\left(p\alpha 2^{\alpha }\Gamma \left(\alpha \right)\right)}^{1/\alpha }$$

is used.

For large $p$ values, when $y_{01}>4.4\alpha +6.0$,

$$y_{03}=-2[ln(1-p)-(\alpha -1)ln\left(\frac{1}{2}{y}_{01}\right)+ln\left(\Gamma \left(\alpha \right)\right)]$$

is found to be a better starting value than $y_{01}$.

For small $\alpha$ $(\alpha \le 0.16)$, $p$ is expressed in terms of an approximation to the exponential integral and $y_{04}$ is found by Newton–Raphson iterations.

Seven terms of the Taylor series are used to refine the starting approximation, repeating the process if necessary until the required accuracy is obtained.

References

Best, D J and Roberts, D E, 1975, Algorithm AS 91. The percentage points of the ${\chi }^2$ distribution, Appl. Statist. (24), 385–388