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: $l (ll)$ – double array: $λ_{i}$ , the parameter of the Poisson distribution with $λ_{i} = l (j)$ , $j = ((i - 1) mod ll) + 1$ , for $i = 1, 2, \dots, \max (ll, lk)$ .

Constraint: $0.0 < l (j) \leq 10^{6}$ , for $j = 1, 2, \dots, ll$ .
2: $k (lk)$ – int64int32nag_int array: $k_{i}$ , the integer which defines the required probabilities with $k_{i} = k (j)$ , $j = ((i - 1) mod lk) + 1$ .

Constraint: $k (j) \geq 0$ , for $j = 1, 2, \dots, lk$ .

Optional Input Parameters

1: $ll$ – int64int32nag_int scalar: Default: the dimension of the array l.
The length of the array l

Constraint: $ll > 0$ .
2: $lk$ – 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 (ll, lk)

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

, the lower tail probabilities.

2: $pgtk (:)$ – double array

The dimension of the array pgtk will be

\max (ll, lk)

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

, the upper tail probabilities.

3: $peqk (:)$ – double array

The dimension of the array peqk will be

\max (ll, lk)

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

, the point probabilities.

4: $ivalid (:)$ – int64int32nag_int array

The dimension of the array ivalid will be

\max (ll, lk)

ivalid (i)

indicates any errors with the input arguments, with

$ivalid (i) = 0$

No error.

$ivalid (i) = 1$

On entry,

λ_{i} \leq 0.0

$ivalid (i) = 2$

On entry,

k_{i} < 0

$ivalid (i) = 3$

On entry,

λ_{i} > 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 l or k was invalid.
Check ivalid for more information.

$ifail = 2$: Constraint: $ll > 0$ .

$ifail = 3$: 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_poisson_vector (g01sk) to calculate each probability depends on

λ_{i}

and

k_{i}

. For given

λ_{i}

, the time is greatest when

k_{i} \approx λ_{i}

, and is then approximately proportional to

\sqrt{λ_{i}}

Example

This example reads a vector of values for

λ

and

k

, and prints the corresponding probabilities.

Open in the MATLAB editor: g01sk_example

function g01sk_example


fprintf('g01sk example results\n\n');

rlamda = [     0.750;  9.200; 34.000; 175.000];
k      = [int64(3); 12;     25;     175];

[plek, pgtk, peqk, ivalid, ifail] = ...
  g01sk(rlamda, k);

fprintf('    rlamda     k     plek      pgtk      peqk\n');
lrlamda = numel(rlamda);
lk      = numel(k);
len     = max ([lrlamda, lk]);
for i=0:len-1
  fprintf('%10.3f%6d%10.5f%10.5f%10.5f\n', rlamda(mod(i,lrlamda)+1), ...
          k(mod(i,lk)+1), plek(i+1), pgtk(i+1), peqk(i+1));
end

g01sk example results

    rlamda     k     plek      pgtk      peqk
     0.750     3   0.99271   0.00729   0.03321
     9.200    12   0.86074   0.13926   0.07755
    34.000    25   0.06736   0.93264   0.02140
   175.000   175   0.52009   0.47991   0.03014

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

Chapter Contents

Chapter Introduction

NAG Toolbox