NAG FL Interface
g01skf (prob_​poisson_​vector)

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1 Purpose

g01skf returns a number of the lower tail, upper tail and point probabilities for the Poisson distribution.

2 Specification

Fortran Interface
Subroutine g01skf ( ll, l, lk, k, plek, pgtk, peqk, ivalid, ifail)
Integer, Intent (In) :: ll, lk, k(lk)
Integer, Intent (Inout) :: ifail
Integer, Intent (Out) :: ivalid(*)
Real (Kind=nag_wp), Intent (In) :: l(ll)
Real (Kind=nag_wp), Intent (Out) :: plek(*), pgtk(*), peqk(*)
C Header Interface
#include <nag.h>
void  g01skf_ (const Integer *ll, const double l[], const Integer *lk, const Integer k[], double plek[], double pgtk[], double peqk[], Integer ivalid[], Integer *ifail)
The routine may be called by the names g01skf or nagf_stat_prob_poisson_vector.

3 Description

Let X = {Xi: i=1 , 2 ,, m } denote a vector of random variables each having a Poisson distribution with parameter λi (>0). Then
Prob{Xi=ki} = e -λi λi ki ki! ,   ki = 0,1,2,  
The mean and variance of each distribution are both equal to λi.
g01skf computes, for given λi and ki the probabilities: Prob{Xiki}, Prob{Xi>ki} and Prob{Xi=ki} using the algorithm described in Knüsel (1986).
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: ll Integer Input
On entry: the length of the array l.
Constraint: ll>0.
2: l(ll) Real (Kind=nag_wp) array Input
On entry: λi, the parameter of the Poisson distribution with λi=l(j), j=((i-1) mod ll)+1, for i=1,2,,max(ll,lk).
Constraint: 0.0<l(j)106, for j=1,2,,ll.
3: lk Integer Input
On entry: the length of the array k.
Constraint: lk>0.
4: 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: k(j)0, for j=1,2,,lk.
5: plek(*) Real (Kind=nag_wp) array Output
Note: the dimension of the array plek must be at least max(ll,lk).
On exit: Prob{Xiki} , the lower tail probabilities.
6: pgtk(*) Real (Kind=nag_wp) array Output
Note: the dimension of the array pgtk must be at least max(ll,lk).
On exit: Prob{Xi>ki} , the upper tail probabilities.
7: peqk(*) Real (Kind=nag_wp) array Output
Note: the dimension of the array peqk must be at least max(ll,lk).
On exit: Prob{Xi=ki} , the point probabilities.
8: ivalid(*) Integer array Output
Note: the dimension of the array ivalid must be at least max(ll,lk).
On exit: ivalid(i) indicates any errors with the input arguments, with
ivalid(i)=0
No error.
ivalid(i)=1
On entry, λi0.0.
ivalid(i)=2
On entry, ki<0.
ivalid(i)=3
On entry, λi>106.
9: 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 l or k was invalid.
Check ivalid for more information.
ifail=2
On entry, array size=value.
Constraint: ll>0.
ifail=3
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 (provided that the results do not underflow to zero).

8 Parallelism and Performance

g01skf is not threaded in any implementation.

9 Further Comments

The time taken by g01skf to calculate each probability depends on λi and ki. For given λi, the time is greatest when kiλi, and is then approximately proportional to λi.

10 Example

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

10.1 Program Text

Program Text (g01skfe.f90)

10.2 Program Data

Program Data (g01skfe.d)

10.3 Program Results

Program Results (g01skfe.r)