NAG Library Function Document

nag_deviates_beta (g01fec)

+− 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_deviates_beta (g01fec) returns the deviate associated with the given lower tail probability of the beta distribution.

2 Specification

#include <nag.h>

#include <nagg01.h>

double

nag_deviates_beta (double p, double a, double b, double tol, NagError *fail)

3 Description

The deviate,

β_{p}

, associated with the lower tail probability,

p

, of the beta distribution with parameters

a

and

b

is defined as the solution to

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

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

An initial approximation,

β_{0}

, to

β_{p}

is found (see Cran et al. (1977)), and the Newton–Raphson iteration

β_{i} = β_{i - 1} - \frac{f (β_{i - 1})}{f^{'} (β_{i - 1})},

where

f (β) = P (B \leq β : a, b) - p

is used, with modifications to ensure that

β

remains in the range

(0, 1)

4 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

5 Arguments

1: p – doubleInput: On entry: $p$ , the lower tail probability from the required beta distribution.
Constraint: $0.0 \leq p \leq 1.0$ .
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: tol – doubleInput: On entry: the relative accuracy required by you in the result. If nag_deviates_beta (g01fec) 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.
5: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

On any of the error conditions listed below except $fail . code =$ NE_RES_NOT_ACC or NE_SOL_NOT_CONV nag_deviates_beta (g01fec) returns $0.0$ .
NE_ALLOC_FAIL: Dynamic memory allocation failed.
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_REAL_ARG_GT: On entry, $a = ⟨value⟩$ and $b = ⟨value⟩$ .
Constraint: $a \leq 10^{6}$ .
On entry, $a = ⟨value⟩$ and $b = ⟨value⟩$ .
Constraint: $b \leq 10^{6}$ .
On entry, $p = ⟨value⟩$ .
Constraint: $p \leq 1.0$ .
NE_REAL_ARG_LE: On entry, $a = ⟨value⟩$ and $b = ⟨value⟩$ .
Constraint: $a > 0.0$ .
On entry, $a = ⟨value⟩$ and $b = ⟨value⟩$ .
Constraint: $b > 0.0$ .
NE_REAL_ARG_LT: On entry, $p = ⟨value⟩$ .
Constraint: $p \geq 0.0$ .
NE_RES_NOT_ACC: The requested accuracy has not been achieved. Use a larger value of tol. There is doubt concerning the accuracy of the computed result. $100$ iterations of the Newton–Raphson method have been performed without satisfying the accuracy criterion (see Section 9). The result should be a reasonable approximation of the solution.
NE_SOL_NOT_CONV: The solution has failed to converge. However, the result should be a reasonable approximation. Requested accuracy not achieved when calculating beta probability. You should try setting tol larger.

7 Accuracy

The required precision, given by tol, should be achieved in most circumstances.

8 Parallelism and Performance

Not applicable.

9 Further Comments

The typical timing will be several times that of nag_prob_beta_dist (g01eec) and will be very dependent on the input argument values. See nag_prob_beta_dist (g01eec) for further comments on timings.

10 Example

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

NAG Library Function Documentnag_deviates_beta (g01fec)