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

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

inv_cdf_beta_vector returns a number of deviates associated with given probabilities of the beta distribution.

For full information please refer to the NAG Library document for g01te

https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/g01/g01tef.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 $\text{i}=1,2,\dots ,max(\text{ltail},\text{lp},\text{la},\text{lb})$:

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

The lower tail probability, i.e., $p_i=(B_i\le {\beta }_{p_i}:a_i,b_i)$.

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

The upper tail probability, i.e., $p_i=(B_i\ge {\beta }_{p_i}:a_i,b_i)$.

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

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

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

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

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

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

tolfloat, optional

The relative accuracy required by you in the results. If inv_cdf_beta_vector 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

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

${\beta }_{p_i}$, the deviates for the beta 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 ${\beta }_{p_i}$.

$ivalid[i-1]=2$

On entry, $p_i<0.0$, or, $p_i>1.0$.

$ivalid[i-1]=3$

On entry, $a_i\le 0.0$, or, $a_i>{10}^6$, or, $b_i\le 0.0$, or, $b_i>{10}^6$.

$ivalid[i-1]=4$

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

$ivalid[i-1]=5$

Requested accuracy not achieved when calculating the beta probability. The result should be a reasonable approximation to the correct solution.

Raises

NagValueError

(errno $2$)

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

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

(errno $3$)

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

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

(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 $tail$, $p$, $a$, or $b$ was invalid, or the solution failed to converge.

Check $ivalid$ for more information.

Notes

The deviate, ${\beta }_{p_i}$, associated with the lower tail probability, $p_i$, of the beta distribution with parameters $a_i$ and $b_i$ is defined as the solution to

$$(B_i\le {\beta }_{p_i}:a_i,b_i)=p_i=\frac{\Gamma (a_i+b_i)}{\Gamma \left(a_i\right)\Gamma \left(b_i\right)}{\int }_0^{{\beta }_{p_i}}{B}_i^{a_i-1}{(1-B_i)}^{b_i-1}d{B}_i\text{,}0\le {\beta }_{p_i}\le 1\text{;}{a}_i,b_i>0\text{.}$$

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

An initial approximation, ${\beta }_{i0}$, to ${\beta }_{p_i}$ is found (see Cran et al. (1977)), and the Newton–Raphson iteration

$${\beta }_k={\beta }_{k-1}-\frac{f_i\left({\beta }_{k-1}\right)}{f_i^{\prime }\left({\beta }_{k-1}\right)}\text{,}$$

where $f_i\left({\beta }_k\right)=(B_i\le {\beta }_k:a_i,b_i)-p_i$ is used, with modifications to ensure that ${\beta }_k$ remains in the range $(0,1)$.

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

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