NAG Library Routine Document

g01sjf (prob_binomial_vector)

1
Purpose

g01sjf returns a number of the lower tail, upper tail and point probabilities for the binomial distribution.

2
Specification

Fortran Interface
Subroutine g01sjf ( ln, n, lp, p, lk, k, plek, pgtk, peqk, ivalid, ifail)
Integer, Intent (In):: ln, n(ln), lp, lk, k(lk)
Integer, Intent (Inout):: ifail
Integer, Intent (Out):: ivalid(*)
Real (Kind=nag_wp), Intent (In):: p(lp)
Real (Kind=nag_wp), Intent (Out):: plek(*), pgtk(*), peqk(*)
C Header Interface
#include <nagmk26.h>
void  g01sjf_ (const Integer *ln, const Integer n[], const Integer *lp, const double p[], const Integer *lk, const Integer k[], double plek[], double pgtk[], double peqk[], Integer ivalid[], Integer *ifail)

3
Description

Let X = Xi: i=1 , 2 ,, m  denote a vector of random variables each having a binomial distribution with parameters ni and pi (ni0 and 0<pi<1). Then
ProbXi=ki= ni ki piki1-pini-ki,  ki=0,1,,ni.  
The mean of the each distribution is given by nipi and the variance by nipi1-pi.
g01sjf computes, for given ni, pi and ki, the probabilities: ProbXiki, ProbXi>ki and ProbXi=ki using an algorithm similar to that described in Knüsel (1986) for the Poisson distribution.
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

Knüsel L (1986) Computation of the chi-square and Poisson distribution SIAM J. Sci. Statist. Comput. 7 1022–1036

5
Arguments

1:     ln – IntegerInput
On entry: the length of the array n.
Constraint: ln>0.
2:     nln – Integer arrayInput
On entry: ni, the first parameter of the binomial distribution with ni=nj, j=i-1 mod ln+1, for i=1,2,,maxln,lp,lk.
Constraint: nj0, for j=1,2,,ln.
3:     lp – IntegerInput
On entry: the length of the array p.
Constraint: lp>0.
4:     plp – Real (Kind=nag_wp) arrayInput
On entry: pi, the second parameter of the binomial distribution with pi=pj, j=i-1 mod lp+1.
Constraint: 0.0<pj<1.0, for j=1,2,,lp.
5:     lk – IntegerInput
On entry: the length of the array k.
Constraint: lk>0.
6:     klk – Integer arrayInput
On entry: ki, the integer which defines the required probabilities with ki=kj, j=i-1 mod lk+1.
Constraint: 0 ki ni .
7:     plek* – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array plek must be at least maxln,lp,lk.
On exit: Prob Xi ki , the lower tail probabilities.
8:     pgtk* – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array pgtk must be at least maxln,lp,lk.
On exit: Prob Xi > ki , the upper tail probabilities.
9:     peqk* – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array peqk must be at least maxln,lp,lk.
On exit: Prob Xi = ki , the point probabilities.
10:   ivalid* – Integer arrayOutput
Note: the dimension of the array ivalid must be at least maxln,lp,lk.
On exit: ivalidi indicates any errors with the input arguments, with
ivalidi=0
No error.
ivalidi=1
On entry,ni<0.
ivalidi=2
On entry,pi0.0,
orpi1.0.
ivalidi=3
On entry,ki<0,
orki>ni.
ivalidi=4
On entry,ni is too large to be represented exactly as a real number.
ivalidi=5
On entry,the variance (=nipi1-pi) exceeds 106.
11:   ifail – IntegerInput/Output
On entry: ifail must be set to 0, -1 or 1. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1 or 1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this argument, the recommended value is 0. 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:
ifail=1
On entry, at least one value of n, p or k was invalid.
Check ivalid for more information.
ifail=2
On entry, array size=value.
Constraint: ln>0.
ifail=3
On entry, array size=value.
Constraint: lp>0.
ifail=4
On entry, array size=value.
Constraint: lk>0.
ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
ifail=-399
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
ifail=-999
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7
Accuracy

Results are correct to a relative accuracy of at least 10-6 on machines with a precision of 9 or more decimal digits, and to a relative accuracy of at least 10-3 on machines of lower precision (provided that the results do not underflow to zero).

8
Parallelism and Performance

g01sjf is not threaded in any implementation.

9
Further Comments

The time taken by g01sjf to calculate each probability depends on the variance (=nipi1-pi) and on ki. For given variance, the time is greatest when kinipi (=the mean), and is then approximately proportional to the square-root of the variance.

10
Example

This example reads a vector of values for n, p and k, and prints the corresponding probabilities.

10.1
Program Text

Program Text (g01sjfe.f90)

10.2
Program Data

Program Data (g01sjfe.d)

10.3
Program Results

Program Results (g01sjfe.r)