NAG Library Function Document

nag_prob_hypergeom_vector (g01slc)

void	nag_prob_hypergeom_vector (Integer ln, const Integer n[], Integer ll, const Integer l[], Integer lm, const Integer m[], Integer lk, const Integer k[], double plek[], double pgtk[], double peqk[], Integer ivalid[], NagError *fail)

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{array}{l} m_{i} \\ k_{i} \end{array}) (\begin{array}{l} n_{i} - m_{i} \\ l_{i} - k_{i} \end{array})}{(\begin{array}{l} n_{i} \\ l_{i} \end{array})},

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)}

nag_prob_hypergeom_vector (g01slc) 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 function 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 – IntegerInput

On entry: the length of the array n.

Constraint:

\ln > 0

2: n[ln] – const IntegerInput

On entry:

n_{i}

, the parameter of the hypergeometric distribution with

n_{i} = n [j]

j = (i - 1) mod \ln

, for

i = 1, 2, \dots, \max (\ln, ll, lm, lk)

Constraint:

n [j - 1] \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] – const IntegerInput

On entry:

l_{i}

, the parameter of the hypergeometric distribution with

l_{i} = l [j]

j = (i - 1) mod ll

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] – const IntegerInput

On entry:

m_{i}

, the parameter of the hypergeometric distribution with

m_{i} = m [j]

j = (i - 1) mod lm

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] – const IntegerInput

On entry:

k_{i}

, the integer which defines the required probabilities with

k_{i} = k [j]

j = (i - 1) mod lk

Constraint:

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

9: plek[ $\dim$ ] – doubleOutput

Note: the dimension, dim, 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[ $\dim$ ] – doubleOutput

Note: the dimension, dim, 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[ $\dim$ ] – doubleOutput

Note: the dimension, dim, 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[ $\dim$ ] – IntegerOutput

Note: the dimension, dim, of the array ivalid must be at least

\max (\ln, ll, lm, lk)

On exit:

ivalid [i - 1]

indicates any errors with the input arguments, with

$ivalid [i - 1] = 0$

No error.

$ivalid [i - 1] = 1$

On entry,

n_{i} < 0

$ivalid [i - 1] = 2$

On entry,	$l_{i} < 0$ ,
or	$l_{i} > n_{i}$ .

$ivalid [i - 1] = 3$

On entry,	$m_{i} < 0$ ,
or	$m_{i} > n_{i}$ .

$ivalid [i - 1] = 4$

On entry,	$k_{i} < 0$ ,
or	$k_{i} > l_{i}$ ,
or	$k_{i} > m_{i}$ ,
or	$k_{i} < l_{i} + m_{i} - n_{i}$ .

$ivalid [i - 1] = 5$

On entry,

n_{i}

is too large to be represented exactly as a real number.

$ivalid [i - 1] = 6$

On entry,

the variance (see Section 3) exceeds

10^{6}

13: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_ARRAY_SIZE: On entry, $array size = ⟨value⟩$ .
Constraint: $lk > 0$ .
On entry, $array size = ⟨value⟩$ .
Constraint: $ll > 0$ .
On entry, $array size = ⟨value⟩$ .
Constraint: $lm > 0$ .
On entry, $array size = ⟨value⟩$ .
Constraint: $\ln > 0$ .
NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NW_IVALID: 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.

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

Not applicable.

9 Further Comments

The time taken by nag_prob_hypergeom_vector (g01slc) 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.

10 Example

This example reads a vector of values for

n

l

m

and

k

, and prints the corresponding probabilities.

NAG Library Function Documentnag_prob_hypergeom_vector (g01slc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_prob_hypergeom_vector (g01slc)