NAG Library Routine Document

g01skf  (prob_poisson_vector)

 Contents

    1  Purpose
    7  Accuracy

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 nagmk26.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)

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: ProbXiki, ProbXi>ki and ProbXi=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 – IntegerInput
On entry: the length of the array l.
Constraint: ll>0.
2:     lll – Real (Kind=nag_wp) arrayInput
On entry: λi, the parameter of the Poisson distribution with λi=lj, j=i-1 mod ll+1, for i=1,2,,maxll,lk.
Constraint: 0.0<lj106, for j=1,2,,ll.
3:     lk – IntegerInput
On entry: the length of the array k.
Constraint: lk>0.
4:     klk – Integer arrayInput
On entry: ki, the integer which defines the required probabilities with ki=kj, j=i-1 mod lk+1.
Constraint: kj0, for j=1,2,,lk.
5:     plek* – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array plek must be at least maxll,lk.
On exit: Prob Xi ki , the lower tail probabilities.
6:     pgtk* – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array pgtk must be at least maxll,lk.
On exit: Prob Xi > ki , the upper tail probabilities.
7:     peqk* – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array peqk must be at least maxll,lk.
On exit: Prob Xi = ki , the point probabilities.
8:     ivalid* – Integer arrayOutput
Note: the dimension of the array ivalid must be at least maxll,lk.
On exit: ivalidi indicates any errors with the input arguments, with
ivalidi=0
No error.
ivalidi=1
On entry,λi0.0.
ivalidi=2
On entry,ki<0.
ivalidi=3
On entry,λi>106.
9:     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 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 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 (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)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017