NAG Library Function Document

nag_prob_non_central_beta_dist (g01gec)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

nag_prob_non_central_beta_dist (g01gec) returns the probability associated with the lower tail of the noncentral beta distribution.

2 Specification

#include <nag.h>

#include <nagg01.h>

double

nag_prob_non_central_beta_dist (double x, double a, double b, double lambda, double tol, Integer max_iter, NagError *fail)

3 Description

The lower tail probability for the noncentral beta distribution with parameters

a

and

b

and noncentrality parameter

λ

P (B \leq β : a, b; λ)

, is defined by

P (B \leq β : a, b; λ) = \sum_{j = 0}^{\infty} e^{- λ / 2} \frac{(λ / 2)}{j!} P (B \leq β : a, b; 0),

(1)

where

P (B \leq β : a, b; 0) = \frac{Γ (a + b)}{Γ (a) Γ (b)} \int_{0}^{β} B^{a - 1} {(1 - B)}^{b - 1} d B,

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 \leq β : a, b; 0)

for each step of the summation (1).

The algorithm is discussed in Lenth (1987).

4 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

5 Arguments

1: $x$ – doubleInput: On entry: $β$ , the deviate from the beta distribution, for which the probability $P (B \leq β : a, b; λ)$ is to be found.

Constraint: $0.0 \leq x \leq 1.0$ .
2: $a$ – doubleInput: On entry: $a$ , the first parameter of the required beta distribution.

Constraint: $0.0 < a \leq 10^{6}$ .
3: $b$ – doubleInput: On entry: $b$ , the second parameter of the required beta distribution.

Constraint: $0.0 < b \leq 10^{6}$ .
4: $lambda$ – doubleInput: On entry: $λ$ , the noncentrality parameter of the required beta distribution.

Constraint: $0.0 \leq lambda \leq - 2.0 \log (U)$ , where $U$ is the safe range parameter as defined by nag_real_safe_small_number (X02AMC).
5: $tol$ – doubleInput: On entry: the relative accuracy required by you in the results. If nag_prob_non_central_beta_dist (g01gec) is entered with tol greater than or equal to $1.0$ or less than $10 \times machine precision$ (see nag_machine_precision (X02AJC)), then the value of $10 \times machine precision$ is used instead.
See Section 7 for the relationship between tol and max_iter.
6: $max_iter$ – IntegerInput: On entry: the maximum number of iterations that the algorithm should use.
See Section 7 for suggestions as to suitable values for max_iter for different values of the arguments.

Suggested value: $500$ .

Constraint: $max_iter \geq 1$ .
7: $fail$ – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction for further information.
NE_CONV: The solution has failed to converge in $〈value〉$ iterations. Consider increasing max_iter or tol.
NE_INT_ARG_LT: On entry, $max_iter = 〈value〉$ .
Constraint: $max_iter \geq 1$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.
NE_PROB_B_INIT: 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. This error exit is no longer possible but is still documented for historical reasons.
NE_PROB_LIMIT: The probability is too close to 0.0 or 1.0 for the algorithm to be able to calculate the required probability. nag_prob_non_central_beta_dist (g01gec) will return 0.0 or 1.0 as appropriate. This should be a reasonable approximation.
NE_REAL_ARG_CONS: On entry, $a = 〈value〉$ .
Constraint: $0.0 < a \leq 10^{6}$ .

On entry, $b = 〈value〉$ .
Constraint: $0.0 < b \leq 10^{6}$ .

On entry, $lambda = 〈value〉$ .
Constraint: $0.0 \leq lambda \leq - 2.0 \log (U)$ , where $U$ is the safe range parameter as defined by nag_real_safe_small_number (X02AMC).

On entry, $x = 〈value〉$ .
Constraint: $0.0 \leq x \leq 1.0$ .

7 Accuracy

Convergence is theoretically guaranteed whenever

P (Y > max_iter) \leq tol

where

Y

has a Poisson distribution with mean

λ / 2

. Excessive round-off errors are possible when the number of iterations used is high and tol is close to machine precision. See Lenth (1987) for further comments on the error bound.

8 Parallelism and Performance

Not applicable.

9 Further Comments

The central beta probabilities can be obtained by setting

lambda = 0.0

10 Example

This example reads values for several beta distributions and calculates and prints the lower tail probabilities until the end of data is reached.

NAG Library Function Documentnag_prob_non_central_beta_dist (g01gec)