nag_deviates_beta_vector (g01tec) (PDF version)
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g01 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_deviates_beta_vector (g01tec)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_deviates_beta_vector (g01tec) returns a number of deviates associated with given probabilities of the beta distribution.

2  Specification

#include <nag.h>
#include <nagg01.h>
void  nag_deviates_beta_vector (Integer ltail, const Nag_TailProbability tail[], Integer lp, const double p[], Integer la, const double a[], Integer lb, const double b[], double tol, double beta[], Integer ivalid[], NagError *fail)

3  Description

The deviate, βpi, associated with the lower tail probability, pi, of the beta distribution with parameters ai and bi is defined as the solution to
P Bi βpi :ai,bi = pi = Γ ai + bi Γ ai Γ bi 0 βpi Bi ai-1 1-Bi bi-1 d Bi ,   0 β pi 1 ; ​ ai , bi > 0 .
The algorithm is a modified version of the Newton–Raphson method, following closely that of Cran et al. (1977).
An initial approximation, βi0, to βpi is found (see Cran et al. (1977)), and the Newton–Raphson iteration
βk = βk-1 - fi βk-1 fi βk-1 ,
where fi βk = P Bi βk :ai,bi - pi  is used, with modifications to ensure that βk remains in the range 0,1.
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 Section 2.6 in the g01 Chapter Introduction for further information.

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:     ltailIntegerInput
On entry: the length of the array tail.
Constraint: ltail>0.
2:     tail[ltail]const Nag_TailProbabilityInput
On entry: indicates which tail the supplied probabilities represent. For j= i-1 mod ltail , for i=1,2,,maxltail,lp,la,lb:
tail[j]=Nag_LowerTail
The lower tail probability, i.e., pi = P Bi βpi : ai , bi .
tail[j]=Nag_UpperTail
The upper tail probability, i.e., pi = P Bi βpi : ai , bi .
Constraint: tail[j-1]=Nag_LowerTail or Nag_UpperTail, for j=1,2,,ltail.
3:     lpIntegerInput
On entry: the length of the array p.
Constraint: lp>0.
4:     p[lp]const doubleInput
On entry: pi, the probability of the required beta distribution as defined by tail with pi=p[j], j=i-1 mod lp.
Constraint: 0.0p[j-1]1.0, for j=1,2,,lp.
5:     laIntegerInput
On entry: the length of the array a.
Constraint: la>0.
6:     a[la]const doubleInput
On entry: ai, the first parameter of the required beta distribution with ai=a[j], j=i-1 mod la.
Constraint: 0.0<a[j-1]106, for j=1,2,,la.
7:     lbIntegerInput
On entry: the length of the array b.
Constraint: lb>0.
8:     b[lb]const doubleInput
On entry: bi, the second parameter of the required beta distribution with bi=b[j], j=i-1 mod lb.
Constraint: 0.0<b[j-1]106, for j=1,2,,lb.
9:     toldoubleInput
On entry: the relative accuracy required by you in the results. If nag_deviates_beta_vector (g01tec) is entered with tol greater than or equal to 1.0 or less than 10×machine precision (see nag_machine_precision (X02AJC)), then the value of 10×machine precision is used instead.
10:   beta[dim]doubleOutput
Note: the dimension, dim, of the array beta must be at least maxltail,lp,la,lb.
On exit: βpi, the deviates for the beta distribution.
11:   ivalid[dim]IntegerOutput
Note: the dimension, dim, of the array ivalid must be at least maxltail,lp,la,lb.
On exit: 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 βpi.
ivalid[i-1]=2
On entry,pi<0.0,
orpi>1.0.
ivalid[i-1]=3
On entry,ai0.0,
orai>106,
orbi0.0,
orbi>106.
ivalid[i-1]=4
The solution has not converged but the result should be a reasonable approximation to the solution.
ivalid[i-1]=5
Requested accuracy not achieved when calculating the beta probability. The result should be a reasonable approximation to the correct solution.
12:   failNagError *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_ARRAY_SIZE
On entry, array size=value.
Constraint: la>0.
On entry, array size=value.
Constraint: lb>0.
On entry, array size=value.
Constraint: lp>0.
On entry, array size=value.
Constraint: ltail>0.
NE_BAD_PARAM
On entry, argument value had an illegal value.
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.
NW_IVALID
On entry, at least one value of tail, p, a, or b was invalid, or the solution failed to converge.
Check ivalid for more information.

7  Accuracy

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

8  Further Comments

The typical timing will be several times that of nag_prob_beta_vector (g01sec) and will be very dependent on the input argument values. See nag_prob_beta_vector (g01sec) for further comments on timings.

9  Example

This example reads lower tail probabilities for several beta distributions and calculates and prints the corresponding deviates.

9.1  Program Text

Program Text (g01tece.c)

9.2  Program Data

Program Data (g01tece.d)

9.3  Program Results

Program Results (g01tece.r)


nag_deviates_beta_vector (g01tec) (PDF version)
g01 Chapter Contents
g01 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012