naginterfaces.library.stat.prob_​gamma_​vector

naginterfaces.library.stat.prob_gamma_vector(tail, g, a, b)

prob_gamma_vector returns a number of lower or upper tail probabilities for the gamma distribution.

For full information please refer to the NAG Library document for g01sf

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g01/g01sff.html

Parameters

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

Indicates whether a lower or upper tail probability is required. For $j=\left(mod(\text{i}-1,\text{ltail})\right)+1-1$, for $\text{i}=1,2,\dots ,max(\text{ltail},\text{lg},\text{la},\text{lb})$:

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

The lower tail probability is returned, i.e., $p_i=(G_i\le g_i:{\alpha }_i,{\beta }_i)$.

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

The upper tail probability is returned, i.e., $p_i=(G_i\ge g_i:{\alpha }_i,{\beta }_i)$.

gfloat, array-like, shape $\left(\text{lg}\right)$

$g_i$, the value of the gamma variate.

afloat, array-like, shape $\left(\text{la}\right)$

The parameter ${\alpha }_i$ of the gamma distribution.

bfloat, array-like, shape $\left(\text{lb}\right)$

The parameter ${\beta }_i$ of the gamma distribution.

Returns

pfloat, ndarray, shape $(max(\text{lg},\text{la}))$

$p_i$, the probabilities of the beta distribution.

ivalidint, ndarray, shape $(max(\text{lg},\text{la}))$

$ivalid[i-1]$ 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, $g_i<0.0$.

$ivalid[i-1]=3$

On entry, ${\alpha }_i\le 0.0$, or, ${\beta }_i\le 0.0$.

$ivalid[i-1]=4$

The solution did not converge in $600$ iterations, see specfun.gamma_incomplete for details. The probability returned should be a reasonable 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: $\text{lg}>0$.

(errno $4$)

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

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

(errno $5$)

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

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

Warns

NagAlgorithmicWarning

(errno $1$)

On entry, at least one value of $g$, $a$, $b$ or $tail$ was invalid, or the solution did not converge.

Check $ivalid$ for more information.

Notes

The lower tail probability for the gamma distribution with parameters ${\alpha }_i$ and ${\beta }_i$, $P(G_i\le g_i)$, is defined by:

$$(G_i\le g_i:{\alpha }_i,{\beta }_i)=\frac{1}{{\beta }_i^{{\alpha }_i}\Gamma \left({\alpha }_i\right)}{\int }_0^{g_i}{G}_i^{{\alpha }_i-1}{e}^{-G_i/{\beta }_i}d{G}_i\text{,}{\alpha }_i>0.0\text{,}{\beta }_i>0.0\text{.}$$

The mean of the distribution is ${\alpha }_i{\beta }_i$ and its variance is ${\alpha }_i{\beta }_i^2$. The transformation $Z_i=\frac{G_i}{{\beta }_i}$ is applied to yield the following incomplete gamma function in normalized form,

$$(G_i\le g_i:{\alpha }_i,{\beta }_i)=(Z_i\le g_i/{\beta }_i:{\alpha }_i,1.0)=\frac{1}{\Gamma \left({\alpha }_i\right)}{\int }_0^{g_i/{\beta }_i}{Z}_i^{{\alpha }_i-1}{e}^{-Z_i}d{Z}_i\text{.}$$

This is then evaluated using specfun.gamma_incomplete.

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

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