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

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth

Parameters

Compulsory Input Parameters

1: $tail (ltail)$ – cell array of strings

Indicates whether the lower or upper tail probabilities are required. For

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

, for

i = 1, 2, \dots, \max (ltail, lx, ldf)

$tail (j) ='L'$: The lower tail probability is returned, i.e., $p_{i} = P (X_{i} \leq x_{i} : ν_{i})$ .
$tail (j) ='U'$: The upper tail probability is returned, i.e., $p_{i} = P (X_{i} \geq x_{i} : ν_{i})$ .

Constraint:

tail (j) ='L'

'U'

, for

j = 1, 2, \dots, ltail

2: $x (lx)$ – double array

x_{i}

, the values of the

χ^{2}

variates with

ν_{i}

degrees of freedom with

x_{i} = x (j)

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

Constraint:

x (j) \geq 0.0

, for

j = 1, 2, \dots, lx

3: $df (ldf)$ – double array

ν_{i}

, the degrees of freedom of the

χ^{2}

-distribution with

ν_{i} = df (j)

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

Constraint:

df (j) > 0.0

, for

j = 1, 2, \dots, ldf

Optional Input Parameters

1: $ltail$ – int64int32nag_int scalar: Default: the dimension of the array tail.
The length of the array tail.

Constraint: $ltail > 0$ .
2: $lx$ – int64int32nag_int scalar: Default: the dimension of the array x.
The length of the array x.

Constraint: $lx > 0$ .
3: $ldf$ – int64int32nag_int scalar: Default: the dimension of the array df.
The length of the array df.

Constraint: $ldf > 0$ .

Output Parameters

1: $p (:)$ – double array

The dimension of the array p will be

\max (ltail, ldf, lx)

p_{i}

, the probabilities for the

χ^{2}

distribution.

2: $ivalid (:)$ – int64int32nag_int array

The dimension of the array ivalid will be

\max (ltail, ldf, lx)

ivalid (i)

indicates any errors with the input arguments, with

$ivalid (i) = 0$

No error.

$ivalid (i) = 1$

On entry,

invalid value supplied in tail when calculating

p_{i}

$ivalid (i) = 2$

On entry,

x_{i} < 0.0

$ivalid (i) = 3$

On entry,

ν_{i} \leq 0.0

$ivalid (i) = 4$

The solution has failed to converge while calculating the gamma variate. The result returned should represent an approximation to the solution.

3: $ifail$ – int64int32nag_int scalar

ifail = 0

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

Error Indicators and Warnings

Note: nag_stat_prob_chisq_vector (g01sc) may return useful information for one or more of the following detected errors or 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 x, df or tail was invalid, or the solution failed to converge.
Check ivalid for more information.

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

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

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

$ifail = - 99$: 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

A relative accuracy of five significant figures is obtained in most cases.

Further Comments

For higher accuracy the transformation described in Description may be used with a direct call to nag_specfun_gamma_incomplete (s14ba).

Example

Values from various

χ^{2}

-distributions are read, the lower tail probabilities calculated, and all these values printed out.

Open in the MATLAB editor: g01sc_example

function g01sc_example


fprintf('g01sc example results\n\n');

x = [8.26; 6.2; 55.76];
df = [20; 7.5; 45];
tail = {'L'};
% calculate probability
[prob, ivalid, ifail] = g01sc( ...
                               tail, x, df);

fprintf('    x      df     prob\n');
lx    = numel(x);
ldf   = numel(df);
ltail = numel(tail);
len   = max ([lx, ldf, ltail]);
for i=0:len-1
  fprintf('%7.3f%8.3f%8.3f\n', x(mod(i,lx)+1), df(mod(i,ldf)+1), prob(i+1));
end

g01sc example results

    x      df     prob
  8.260  20.000   0.010
  6.200   7.500   0.428
 55.760  45.000   0.869

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

Chapter Contents

Chapter Introduction

NAG Toolbox