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 Vectorized Routines in the G01 Chapter Introduction for further information.

References

Knüsel L (1986) Computation of the chi-square and Poisson distribution SIAM J. Sci. Statist. Comput. 7 1022–1036

Parameters

Compulsory Input Parameters

1: $n (\ln)$ – int64int32nag_int array: $n_{i}$ , 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 $j = 1, 2, \dots, \ln$ .
2: $l (ll)$ – int64int32nag_int array: $l_{i}$ , the parameter of the hypergeometric distribution with $l_{i} = l (j)$ , $j = ((i - 1) mod ll) + 1$ .

Constraint: $0 \leq l_{i} \leq n_{i}$ .
3: $m (lm)$ – int64int32nag_int array: $m_{i}$ , the parameter of the hypergeometric distribution with $m_{i} = m (j)$ , $j = ((i - 1) mod lm) + 1$ .

Constraint: $0 \leq m_{i} \leq n_{i}$ .
4: $k (lk)$ – int64int32nag_int array: $k_{i}$ , the integer which defines the required probabilities with $k_{i} = k (j)$ , $j = ((i - 1) mod lk) + 1$ .

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

Optional Input Parameters

1: $\ln$ – int64int32nag_int scalar: Default: the dimension of the array n.
The length of the array n

Constraint: $\ln > 0$ .
2: $ll$ – int64int32nag_int scalar: Default: the dimension of the array l.
The length of the array l

Constraint: $ll > 0$ .
3: $lm$ – int64int32nag_int scalar: Default: the dimension of the array m.
The length of the array m

Constraint: $lm > 0$ .
4: $lk$ – int64int32nag_int scalar: Default: the dimension of the array k.
The length of the array k

Constraint: $lk > 0$ .

Output Parameters

1: $plek (:)$ – double array

The dimension of the array plek will be

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

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

, the lower tail probabilities.

2: $pgtk (:)$ – double array

The dimension of the array pgtk will be

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

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

, the upper tail probabilities.

3: $peqk (:)$ – double array

The dimension of the array peqk will be

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

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

, the point probabilities.

4: $ivalid (:)$ – int64int32nag_int array

The dimension of the array ivalid will be

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

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 Description) exceeds

10^{6}

5: $ifail$ – int64int32nag_int scalar

ifail = 0

unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

W $ifail = 1$: 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.

$ifail = 2$: Constraint: $\ln > 0$ .

$ifail = 3$: Constraint: $ll > 0$ .

$ifail = 4$: Constraint: $lm > 0$ .

$ifail = 5$: Constraint: $lk > 0$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

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).

Further Comments

The time taken by nag_stat_prob_hypergeom_vector (g01sl) to calculate each probability depends on the variance (see Description) 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.

Example

This example reads a vector of values for

n

l

m

and

k

, and prints the corresponding probabilities.

Open in the MATLAB editor: g01sl_example

function g01sl_example


fprintf('g01sl example results\n\n');

n = [int64(10); 40; 155; 1000];
l = [int64( 2); 10;  35;  444];
m = [int64( 5);  3; 122;  500];
k = [int64( 1);  2;  22;  220];

[plek, pgtk, peqk, ivalid, ifail] = ...
  g01sl(n, l, m, k);

fprintf('   n   l   m   k     plek      pgtk      peqk\n');
ln  = numel(n);
ll  = numel(l);
lm  = numel(m);
lk  = numel(k);
len = max ([ln, ll, lm, lk]);
for i=0:len-1
  fprintf('%4d%4d%4d%4d%10.5f%10.5f%10.5f\n', n(mod(i,ln)+1), ...
          l(mod(i,ll)+1), m(mod(i,lm)+1), k(mod(i,lk)+1), plek(i+1), ...
          pgtk(i+1), peqk(i+1));
end

g01sl example results

   n   l   m   k     plek      pgtk      peqk
  10   2   5   1   0.77778   0.22222   0.55556
  40  10   3   2   0.98785   0.01215   0.13664
 155  35 122  22   0.01101   0.98899   0.00779
1000 444 500 220   0.42429   0.57571   0.04913

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox