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

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

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 = \{X_{i} : i = 1, 2, \dots, r\}

denote a vector of random variables having a hypergeometric distribution with parameters

n_{i}

l_{i}

and

m_{i}

. Then

Prob \{X_{i} = k_{i}\} = \frac{(\begin{matrix} m_{i} \\ k_{i} \end{matrix}) (\begin{matrix} n_{i} - m_{i} \\ l_{i} - k_{i} \end{matrix})}{(\begin{matrix} n_{i} \\ l_{i} \end{matrix})},

where

\max (0, l_{i} + m_{i} - n_{i}) \leq k_{i} \leq \min (l_{i}, m_{i})

0 \leq l_{i} \leq n_{i}

and

0 \leq m_{i} \leq n_{i}

The hypergeometric distribution may arise if in a population of size

n_{i}

a number

m_{i}

are marked. From this population a sample of size

l_{i}

is drawn and of these

k_{i}

are observed to be marked.

The mean of the distribution

= \frac{l_{i} m_{i}}{n_{i}}

, and the variance

= \frac{l_{i} m_{i} (n_{i} - l_{i}) (n_{i} - m_{i})}{{n_{i}}^{2} (n_{i} - 1)}

G01SLF computes for given

n_{i}

l_{i}

m_{i}

and

k_{i}

the probabilities:

Prob \{X_{i} \leq k_{i}\}

Prob \{X_{i} > k_{i}\}

and

Prob \{X_{i} = k_{i}\}

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:

n_{i}

, the parameter of the hypergeometric distribution with

n_{i} = N (j)

j = ((i - 1) mod LN) + 1

, for

i = 1, 2, \dots, \max (LN, LL, LM, LK)

Constraint:

N (j) \geq 0

, for

j = 1, 2, \dots, LN

3: LL – INTEGERInput

On entry: the length of the array L

Constraint:

LL > 0

4: L(LL) – INTEGER arrayInput

On entry:

l_{i}

, the parameter of the hypergeometric distribution with

l_{i} = L (j)

j = ((i - 1) mod LL) + 1

Constraint:

0 \leq l_{i} \leq n_{i}

5: LM – INTEGERInput

On entry: the length of the array M

Constraint:

LM > 0

6: M(LM) – INTEGER arrayInput

On entry:

m_{i}

, the parameter of the hypergeometric distribution with

m_{i} = M (j)

j = ((i - 1) mod LM) + 1

Constraint:

0 \leq m_{i} \leq n_{i}

7: LK – INTEGERInput

On entry: the length of the array K

Constraint:

LK > 0

8: K(LK) – INTEGER arrayInput

On entry:

k_{i}

, the integer which defines the required probabilities with

k_{i} = K (j)

j = ((i - 1) mod LK) + 1

Constraint:

\max (0, l_{i} + m_{i} - n_{i}) \leq k_{i} \leq \min (l_{i}, m_{i})

9: PLEK( $*$ ) – REAL (KIND=nag_wp) arrayOutput

Note: the dimension of the array PLEK must be at least

\max (LN, LL, LM, LK)

On exit:

Prob \{X_{i} \leq k_{i}\}

, the lower tail probabilities.

10: PGTK( $*$ ) – REAL (KIND=nag_wp) arrayOutput

Note: the dimension of the array PGTK must be at least

\max (LN, LL, LM, LK)

On exit:

Prob \{X_{i} > k_{i}\}

, the upper tail probabilities.

11: PEQK( $*$ ) – REAL (KIND=nag_wp) arrayOutput

Note: the dimension of the array PEQK must be at least

\max (LN, LL, LM, LK)

On exit:

Prob \{X_{i} = k_{i}\}

, the point probabilities.

12: IVALID( $*$ ) – INTEGER arrayOutput

Note: the dimension of the array IVALID must be at least

\max (LN, LL, LM, LK)

On exit:

IVALID (i)

indicates any errors with the input arguments, with

$IVALID (i) = 0$: No error.
$IVALID (i) = 1$: On entry, $n_{i} < 0$ .
$IVALID (i) = 2$: On entry, $l_{i} < 0$ ,
or $l_{i} > n_{i}$ .
$IVALID (i) = 3$: On entry, $m_{i} < 0$ ,
or $m_{i} > n_{i}$ .
$IVALID (i) = 4$: On entry, $k_{i} < 0$ ,
or $k_{i} > l_{i}$ ,
or $k_{i} > m_{i}$ ,
or $k_{i} < l_{i} + m_{i} - n_{i}$ .
$IVALID (i) = 5$: On entry, $n_{i}$ is too large to be represented exactly as a real number.
$IVALID (i) = 6$: On entry, the variance (see Section 3) exceeds $10^{6}$ .

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

- 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

k_{i}

. For given variance, the time is greatest when

k_{i} \approx l_{i} m_{i} / n_{i}

(

=

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.

NAG Library Routine DocumentG01SLF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

G01SLF