NAG FL Interface
g01tef (inv_​cdf_​beta_​vector)

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1 Purpose

g01tef returns a number of deviates associated with given probabilities of the beta distribution.

2 Specification

Fortran Interface
Subroutine g01tef ( ltail, tail, lp, p, la, a, lb, b, tol, beta, ivalid, ifail)
Integer, Intent (In) :: ltail, lp, la, lb
Integer, Intent (Inout) :: ifail
Integer, Intent (Out) :: ivalid(*)
Real (Kind=nag_wp), Intent (In) :: p(lp), a(la), b(lb), tol
Real (Kind=nag_wp), Intent (Out) :: beta(*)
Character (1), Intent (In) :: tail(ltail)
C Header Interface
#include <nag.h>
void  g01tef_ (const Integer *ltail, const char tail[], const Integer *lp, const double p[], const Integer *la, const double a[], const Integer *lb, const double b[], const double *tol, double beta[], Integer ivalid[], Integer *ifail, const Charlen length_tail)
The routine may be called by the names g01tef or nagf_stat_inv_cdf_beta_vector.

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 routine 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: ltail Integer Input
On entry: the length of the array tail.
Constraint: ltail>0.
2: tail(ltail) Character(1) array Input
On entry: indicates which tail the supplied probabilities represent. For j= ((i-1) mod ltail) +1 , for i=1,2,,max(ltail,lp,la,lb):
tail(j)='L'
The lower tail probability, i.e., pi = P( Bi βpi : ai , bi ) .
tail(j)='U'
The upper tail probability, i.e., pi = P( Bi βpi : ai , bi ) .
Constraint: tail(j)='L' or 'U', for j=1,2,,ltail.
3: lp Integer Input
On entry: the length of the array p.
Constraint: lp>0.
4: p(lp) Real (Kind=nag_wp) array Input
On entry: pi, the probability of the required beta distribution as defined by tail with pi=p(j), j=((i-1) mod lp)+1.
Constraint: 0.0p(j)1.0, for j=1,2,,lp.
5: la Integer Input
On entry: the length of the array a.
Constraint: la>0.
6: a(la) Real (Kind=nag_wp) array Input
On entry: ai, the first parameter of the required beta distribution with ai=a(j), j=((i-1) mod la)+1.
Constraint: 0.0<a(j)106, for j=1,2,,la.
7: lb Integer Input
On entry: the length of the array b.
Constraint: lb>0.
8: b(lb) Real (Kind=nag_wp) array Input
On entry: bi, the second parameter of the required beta distribution with bi=b(j), j=((i-1) mod lb)+1.
Constraint: 0.0<b(j)106, for j=1,2,,lb.
9: tol Real (Kind=nag_wp) Input
On entry: the relative accuracy required by you in the results. If g01tef is entered with tol greater than or equal to 1.0 or less than 10×machine precision (see x02ajf), the value of 10×machine precision is used instead.
10: beta(*) Real (Kind=nag_wp) array Output
Note: the dimension of the array beta must be at least max(ltail,lp,la,lb).
On exit: βpi, the deviates for the beta distribution.
11: ivalid(*) Integer array Output
Note: the dimension of the array ivalid must be at least max(ltail,lp,la,lb).
On exit: ivalid(i) indicates any errors with the input arguments, with
ivalid(i)=0
No error.
ivalid(i)=1
On entry, invalid value supplied in tail when calculating βpi.
ivalid(i)=2
On entry, pi<0.0, or, pi>1.0.
ivalid(i)=3
On entry, ai0.0, or, ai>106, or, bi0.0, or, bi>106.
ivalid(i)=4
The solution has not converged but the result should be a reasonable approximation to the solution.
ivalid(i)=5
Requested accuracy not achieved when calculating the beta probability. The result should be a reasonable approximation to the correct solution.
12: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value −1 is recommended since useful values can be provided in some output arguments even when ifail0 on exit. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry ifail=0 or −1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
Note: in some cases g01tef may return useful information.
ifail=1
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.
ifail=2
On entry, array size=value.
Constraint: ltail>0.
ifail=3
On entry, array size=value.
Constraint: lp>0.
ifail=4
On entry, array size=value.
Constraint: la>0.
ifail=5
On entry, array size=value.
Constraint: lb>0.
ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
ifail=-399
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
ifail=-999
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

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

8 Parallelism and Performance

g01tef is not threaded in any implementation.

9 Further Comments

The typical timing will be several times that of g01sef and will be very dependent on the input argument values. See g01sef for further comments on timings.

10 Example

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

10.1 Program Text

Program Text (g01tefe.f90)

10.2 Program Data

Program Data (g01tefe.d)

10.3 Program Results

Program Results (g01tefe.r)