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The inverse of the incomplete beta integral

Citation for published version (APA):

Newby, M. J. (1991). The inverse of the incomplete beta integral. (TH Eindhoven. THE/BDK/ORS, Vakgroep ORS : rapporten; Vol. 9107). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1991

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ARW

16

BDK

9107

Technische Universiteit Eindhoven Faculteit Technische Bedrijfskunde

THE INVERSE OF THE INCOMPLETE BET A INTEGRAL

Dr. M.J. Newby

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THE INVERSE OF THE INCOMPLETE BETA INTEGRAL

Martin Newbyt,

Faculty of Industrial Engineering and Management, Eindhoven University of Technology,

and

Frits Philips Institute for Quality Management, Eindhoven

Keywords: beta integral; F -distribution; beta distribution; inverse beta function

LANGUAGE FORTRAN 77

DESCRIPTION AND PURPOSE

For a given probability Q:e[O,l] and m>O, n>O the subprogram returns x(lte[O,l}, the percentile of the beta distribution satisfying

X(lt 1

J

m-l n-l - - - U (1 - u) du 8(m,n) o NUMERICAL METHOD

Algorithm AS 64/AS 109 (Majumder and Chattacharjee, 1973; Cran, et al., 1977) uses an approximation to determine an initial value for X(lt and thereafter a modified Newton-Raphson method to produce the required accuracy. The modifications are required to ensure that the returned value lies in the appropriate range. When, for example, m>1 and n<1 the convergence is very slow because the iteration tries to push the x values outside the interval [0,1]. These difficulties are remarkably easily resolved and a clearer and simpler algorithm obtained by remarking that the integral is a monotone increasing function of x for xe[O,l]. Because the support of the beta function is [0,1] repeated

t Address for correspondence: Facu1ty of Industrial Engineering and Management, Eindhoven University of Technology, Den Dolech 2, PO Box 513, 5600 MB Eindhoven, The Netherlands.

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bisection converges rapidly to the root without the need for any special precautions. Thus although the algorithm is not uniformly better than AS 64/AS 109 it is reliable and cannot fail. No initial approximations to the solution are needed and it is expressed entirely in terms of the incomplete beta integral.

STRUCTURE

REAL FUNCTION BTAINV(M,N,ALPHA,B,IF AlL)

Formal parameters

M Real

N Real

ALPHA Real

B Real

input: parameter m of the beta integral input: parameter n of the beta integral input: the probability level

input/output: the logarithm of the beta function. If B>O it is used

as the value of B(p,q). If BsO the value of B(p,q) is evaluated within the BETAIN subprogram and is available through B for later use [FAlL Integer output: error flag, IF AlL=O indicates success

IF AlL=l PsO or QsO

AUXILIARY ALGORITHMS

BTAINV uses the function BETAIN(X,P,Q,B,IFAlL) to evaluate the incomplete beta function, BET AIN in turn requires BET AO and the logarithm of the gamma function.

REFERENCES

Cran, C. W., Martin, K. J. and Thomas, C. E. (1977) Remark AS R19 and algorithm AS 109,

Applied Statistics, 26, 111-114.

Majumder, K. L. and Bhattacharjee, G. P. (1973) Algorithm AS 63. The Incomplete Beta Integral, Applied Statistics, 22, 409-411.

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REAL FUNCTION BTAINV(M,N,ALPHA,B,IFAIL) IMPLICIT REAL (A-H,O-Z)

C*****************************************************

***************** C C FUNCTION BTAINV(M,N,ALPHA,B,IFAIL) C

C*****************************************************

***************** REAL P,ALPHA,XO,X1,X2 .,B1, M,N, 1 ZERO,ONE,HALF,TOL INTEGER NOUGHT,UNITY,TWO,IFAIL PARAMETER ( ZERO == O.OEO,

1 ONE == 1.0EO, 2 TWO == 2, 3 HALF = 0.5EO, 4 UNITY

=

1, 5 NOUGHT = 0,

C*****************************************************

***************** C

C NSTEP IS SUCH THAT 2**( -NSTEP)<=TOL.

C I.E X IS IN [X-.5*TOL,X+.5*TOL]. ABOUT 3.3 STEPS PER DECIMAL PLACE C ARE NEEDED. 17 STEPS GIVE 5DP, 20 GIVE 6DP AND 23 7DP

C

C*****************************************************

***************** 6 7 P=ALPHA NSTEP:::; 23, TOL :::; 1.0E-5

C*****************************************************

***************** C

C DEAL WITH TRMAL CASES AND ARGUMENT CHECKING C

C*****************************************************

***************** IF AIL :::; NOUGHT

IF ( P .LT. ZERO .OR. P .GT. ONE ) THEN IF AIL

=

UNITY

GO TO 1000

ELSEIF ( M .LE. ZERO .OR. N .LE. ZERO ) THEN IFAIL = TWO

GO TO 1000

ELSEIF ( P .EQ. ZERO ) THEN BTAINV

=

ZERO

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ELSEIF ( P .EQ. ONE ) THEN BTAINV = ONE

CO TO 1000

ELSEIF ( M .EQ. ONE ) THEN

BTAINV = ONE - (ONE - P)**(ONE/N) CO TO 1000

ELSEIF ( N .EQ. ONE ) THEN BTAINV = P**(ONE/M) CO TO 1000

ELSE

C**********************************************************************

C

C FIND THE INVERSE ONLY FOR THOSE P THAT ARE NOT ZERO OR ONE C USE SIMPLE BISECTION. IF B <= 0 THE VALUE IS RETURNED BY THE C FIRST EVALUATION OF BETAIN AND THEN CARRIED FORWARD

C C********************************************************************** XO = ZERO X2 = ONE DO 10,I=I,NSTEP Xl = HALF*(XO

+

X2) Bl = BETAIN(Xl,M,N,B,IFAULT) IF ( P .CT. Bl) THEN XO = Xl ELSE X2 = Xl ENDIF 10 CONTINUE BTAINV = Xl ENDIF 1000 CONTINUE RETURN END

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