naginterfaces.library.stat.prob_​beta_​vector

naginterfaces.library.stat.prob_beta_vector(tail, beta, a, b)

prob_beta_vector computes a number of lower or upper tail probabilities for the beta distribution.

For full information please refer to the NAG Library document for g01se

https://support.nag.com/numeric/nl/nagdoc_30.1/flhtml/g01/g01sef.html

Parameters

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

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

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

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

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

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

betafloat, array-like, shape $\left(\text{lbeta}\right)$

${\beta }_i$, the value of the beta variate.

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.

Returns

pfloat, ndarray, shape $(max(\text{ltail},\text{lbeta}))$

$p_i$, the probabilities for the beta distribution.

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

$ivalid[i-1]$ 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, ${\beta }_i<0.0$, or, ${\beta }_i>1.0$.

$ivalid[i-1]=3$

On entry, $a_i\le 0.0$, or, $b_i\le 0.0$.

Raises

NagValueError

(errno $2$)

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

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

(errno $3$)

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

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

(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 $beta$, $a$, $b$ or $tail$ was invalid.

Check $ivalid$ for more information.

Notes

The lower tail probability, $(B_i\le {\beta }_i:a_i,b_i)$ is defined by

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

The function $I_{{\beta }_i}(a_i,b_i)$, also known as the incomplete beta function is calculated using specfun.beta_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

NIST Digital Library of Mathematical Functions

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

Majumder, K L and Bhattacharjee, G P, 1973, Algorithm AS 63. The incomplete beta integral, Appl. Statist. (22), 409–411