G01SLF (PDF version)
G01 Chapter Contents
G01 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

G01SLF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

G01SLF returns a number of the lower tail, upper tail and point probabilities for the hypergeometric distribution.

2  Specification

SUBROUTINE G01SLF ( LN, N, LL, L, LM, M, LK, K, PLEK, PGTK, PEQK, IVALID, IFAIL)
INTEGER  LN, N(LN), LL, L(LL), LM, M(LM), LK, K(LK), IVALID(*), IFAIL
REAL (KIND=nag_wp)  PLEK(*), PGTK(*), PEQK(*)

3  Description

Let X = Xi: i=1 , 2 ,, r  denote a vector of random variables having a hypergeometric distribution with parameters ni, li and mi. Then
Prob Xi = ki = mi ki ni - mi li - ki ni li ,
where max0, li + mi - ni ki minli,mi , 0lini and 0mini.
The hypergeometric distribution may arise if in a population of size ni a number mi are marked. From this population a sample of size li is drawn and of these ki are observed to be marked.
The mean of the distribution = limi ni , and the variance = limini-lini-mi ni2ni-1 .
G01SLF computes for given ni, li, mi 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 parameters 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  Parameters

1:     LN – INTEGERInput
On entry: the length of the array N
Constraint: LN>0.
2:     N(LN) – INTEGER arrayInput
On entry: ni, the parameter of the hypergeometric distribution with ni=Nj, j=i-1 mod LN+1, for i=1,2,,maxLN,LL,LM,LK.
Constraint: Nj0, for j=1,2,,LN.
3:     LL – INTEGERInput
On entry: the length of the array L
Constraint: LL>0.
4:     L(LL) – INTEGER arrayInput
On entry: li, the parameter of the hypergeometric distribution with li=Lj, j=i-1 mod LL+1.
Constraint: 0 li ni .
5:     LM – INTEGERInput
On entry: the length of the array M
Constraint: LM>0.
6:     M(LM) – INTEGER arrayInput
On entry: mi, the parameter of the hypergeometric distribution with mi=Mj, j=i-1 mod LM+1.
Constraint: 0 mi ni .
7:     LK – INTEGERInput
On entry: the length of the array K
Constraint: LK>0.
8:     K(LK) – INTEGER arrayInput
On entry: ki, the integer which defines the required probabilities with ki=Kj, j=i-1 mod LK+1.
Constraint: max0, li + mi - ni ki minli,mi .
9:     PLEK(*) – REAL (KIND=nag_wp) arrayOutput
Note: the dimension of the array PLEK must be at least maxLN,LL,LM,LK.
On exit: Prob Xi ki , the lower tail probabilities.
10:   PGTK(*) – REAL (KIND=nag_wp) arrayOutput
Note: the dimension of the array PGTK must be at least maxLN,LL,LM,LK.
On exit: Prob Xi > ki , the upper tail probabilities.
11:   PEQK(*) – REAL (KIND=nag_wp) arrayOutput
Note: the dimension of the array PEQK must be at least maxLN,LL,LM,LK.
On exit: Prob Xi = ki , the point probabilities.
12:   IVALID(*) – INTEGER arrayOutput
Note: the dimension of the array IVALID must be at least maxLN,LL,LM,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,li<0,
orli>ni.
IVALIDi=3
On entry,mi<0,
ormi>ni.
IVALIDi=4
On entry,ki<0,
orki>li,
orki>mi,
orki<li+mi-ni.
IVALIDi=5
On entry,ni is too large to be represented exactly as a real number.
IVALIDi=6
On entry,the variance (see Section 3) exceeds 106.
13:   IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to 0, -1​ or ​1. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction 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 parameter, 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, L, M or K was invalid, or the variance was too large.
Check IVALID for more information.
IFAIL=2
On entry, array size=value.
Constraint: LN>0.
IFAIL=3
On entry, array size=value.
Constraint: LL>0.
IFAIL=4
On entry, array size=value.
Constraint: LM>0.
IFAIL=5
On entry, array size=value.
Constraint: LK>0.
IFAIL=-999
Dynamic memory allocation failed.

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  Further Comments

The time taken by G01SLF to calculate each probability depends on the variance (see Section 3) and on ki. For given variance, the time is greatest when kilimi/ni (= the mean), and is then approximately proportional to the square-root of the variance.

9  Example

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

9.1  Program Text

Program Text (g01slfe.f90)

9.2  Program Data

Program Data (g01slfe.d)

9.3  Program Results

Program Results (g01slfe.r)


G01SLF (PDF version)
G01 Chapter Contents
G01 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012