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.
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.
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.
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 argument 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)