NAG FL Interface
g01sjf (prob_​binomial_​vector)

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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 <nag.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)
The routine may be called by the names g01sjf or nagf_stat_prob_binomial_vector.

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
Prob{Xi=ki}=( ni ki ) piki(1-pi)ni-ki,  ki=0,1,,ni.  
The mean of the each distribution is given by nipi and the variance by nipi(1-pi).
g01sjf computes, for given ni, pi and ki, the probabilities: Prob{Xiki}, Prob{Xi>ki} and Prob{Xi=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 Integer Input
On entry: the length of the array n.
Constraint: ln>0.
2: n(ln) Integer array Input
On entry: ni, the first parameter of the binomial distribution with ni=n(j), j=((i-1) mod ln)+1, for i=1,2,,max(ln,lp,lk).
Constraint: n(j)0, for j=1,2,,ln.
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 second parameter of the binomial distribution with pi=p(j), j=((i-1) mod lp)+1.
Constraint: 0.0<p(j)<1.0, for j=1,2,,lp.
5: lk Integer Input
On entry: the length of the array k.
Constraint: lk>0.
6: k(lk) Integer array Input
On entry: ki, the integer which defines the required probabilities with ki=k(j), j=((i-1) mod lk)+1.
Constraint: 0 ki ni .
7: plek(*) Real (Kind=nag_wp) array Output
Note: the dimension of the array plek must be at least max(ln,lp,lk).
On exit: Prob{Xiki} , the lower tail probabilities.
8: pgtk(*) Real (Kind=nag_wp) array Output
Note: the dimension of the array pgtk must be at least max(ln,lp,lk).
On exit: Prob{Xi>ki} , the upper tail probabilities.
9: peqk(*) Real (Kind=nag_wp) array Output
Note: the dimension of the array peqk must be at least max(ln,lp,lk).
On exit: Prob{Xi=ki} , the point probabilities.
10: ivalid(*) Integer array Output
Note: the dimension of the array ivalid must be at least max(ln,lp,lk).
On exit: ivalid(i) indicates any errors with the input arguments, with
ivalid(i)=0
No error.
ivalid(i)=1
On entry, ni<0.
ivalid(i)=2
On entry, pi0.0, or, pi1.0.
ivalid(i)=3
On entry, ki<0, or, ki>ni.
ivalid(i)=4
On entry, ni is too large to be represented exactly as a real number.
ivalid(i)=5
On entry, the variance (=nipi(1-pi)) exceeds 106.
11: 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 0 is recommended. 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 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

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 (=nipi(1-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)