naginterfaces.library.stat.prob_​beta_​noncentral

naginterfaces.library.stat.prob_beta_noncentral(x, a, b, rlamda, tol=0.0, maxit=500)

prob_beta_noncentral returns the probability associated with the lower tail of the noncentral beta distribution.

For full information please refer to the NAG Library document for g01ge

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/g01/g01gef.html

Parameters

xfloat

$\beta$, the deviate from the beta distribution, for which the probability $P(B\le \beta :a,b\text{;}\lambda )$ is to be found.

afloat

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

bfloat

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

rlamdafloat

$\lambda$, the noncentrality parameter of the required beta distribution.

tolfloat, optional

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

See Accuracy for the relationship between $tol$ and $maxit$.

maxitint, optional

The maximum number of iterations that the algorithm should use.

See Accuracy for suggestions as to suitable values for $maxit$ for different values of the arguments.

Returns

pfloat

The probability associated with the lower tail of the noncentral beta distribution.

Raises

NagValueError

(errno $1$)

On entry, $b=⟨value⟩$.

Constraint: $0.0<b\le {10}^6$.

(errno $1$)

On entry, $maxit=⟨value⟩$.

Constraint: $maxit\ge 1$.

(errno $1$)

On entry, $a=⟨value⟩$.

Constraint: $0.0<a\le {10}^6$.

(errno $1$)

On entry, $x=⟨value⟩$.

Constraint: $0.0\le x\le 1.0$.

(errno $1$)

On entry, $rlamda=⟨value⟩$.

Constraint: $0.0\le rlamda\le -2.0\log \left(U\right)$, where $U$ is the safe range parameter as defined by machine.real_safe.

Warns

NagAlgorithmicWarning

(errno $2$)

The solution has failed to converge in $⟨value⟩$ iterations. Consider increasing $maxit$ or $tol$. The returned value will be an approximation to the correct value.

(errno $3$)

The probability is too close to $0.0$ or $1.0$ for the algorithm to be able to calculate the required probability. prob_beta_noncentral will return $0.0$ or $1.0$ as appropriate. This should be a reasonable approximation.

(errno $4$)

The required accuracy was not achieved when calculating the initial value of the beta distribution. You should try a larger value of $tol$. The returned value will be an approximation to the correct value.

Notes

The lower tail probability for the noncentral beta distribution with parameters $a$ and $b$ and noncentrality parameter $\lambda$, $P(B\le \beta :a,b\text{;}\lambda )$, is defined by

$$P(B\le \beta :a,b\text{;}\lambda )=\sum _{j=0}^{\infty }{e}^{-\lambda /2}\frac{\left(\lambda /2\right)}{j!}P(B\le \beta :a,b\text{;}0)\text{,}$$

where

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

which is the central beta probability function or incomplete beta function.

Recurrence relationships given in Abramowitz and Stegun (1972) are used to compute the values of $P(B\le \beta :a,b\text{;}0)$ for each step of the summation (1).

The algorithm is discussed in Lenth (1987).

References

Abramowitz, M and Stegun, I A, 1972, Handbook of Mathematical Functions, (3rd Edition), Dover Publications

Lenth, R V, 1987, Algorithm AS 226: Computing noncentral beta probabilities, Appl. Statist. (36), 241–244