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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_stat_prob_chisq (g01ec)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_stat_prob_chisq (g01ec) returns the lower or upper tail probability for the χ2-distribution with real degrees of freedom.


[result, ifail] = g01ec(x, df, 'tail', tail)
[result, ifail] = nag_stat_prob_chisq(x, df, 'tail', tail)
Note: the interface to this routine has changed since earlier releases of the toolbox:
At Mark 23: tail was made optional (default 'L')


The lower tail probability for the χ2-distribution with ν degrees of freedom, PXx:ν is defined by:
PXx:ν=12ν/2Γν/2 0.0xXν/2-1e-X/2dX,  x0,ν>0.  
To calculate PXx:ν a transformation of a gamma distribution is employed, i.e., a χ2-distribution with ν degrees of freedom is equal to a gamma distribution with scale parameter 2 and shape parameter ν/2.


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


Compulsory Input Parameters

1:     x – double scalar
x, the value of the χ2 variate with ν degrees of freedom.
Constraint: x0.0.
2:     df – double scalar
ν, the degrees of freedom of the χ2-distribution.
Constraint: df>0.0.

Optional Input Parameters

1:     tail – string (length ≥ 1)
Default: 'L'
Indicates whether the upper or lower tail probability is required.
The lower tail probability is returned, i.e., PXx:ν.
The upper tail probability is returned, i.e., PXx:ν.
Constraint: tail='L' or 'U'.

Output Parameters

1:     result – double scalar
The result of the function.
2:     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 (g01ec) may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the function:
If ifail=1, 2 or 3 on exit, then nag_stat_prob_chisq (g01ec) returns 0.0.
On entry,tail'L' or 'U'.
On entry,x<0.0.
On entry,df0.0.
The solution has failed to converge while calculating the gamma variate. The result returned should represent an approximation to the solution.
An unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


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).


Values from various χ2-distributions are read, the lower tail probabilities calculated, and all these values printed out, until the end of data is reached.
function g01ec_example

fprintf('g01ec example results\n\n');

% Lower tail probabilities for Chi^2 distribution
x  = [ 8.26; 6.2; 55.76];
df = [20.00; 7.5; 45.00];
tail = {'Lower'; 'Lower'; 'Lower';};

fprintf('  Tail    x      df    probability\n');
for j = 1:numel(x);

  [p, ifail] = g01ec( ...
                      x(j), df(j), 'tail', tail{j});

  fprintf('%4s%8.3f%8.1f%12.4f\n',tail{j}(1), x(j), df(j), p);

g01ec example results

  Tail    x      df    probability
   L   8.260    20.0      0.0100
   L   6.200     7.5      0.4279
   L  55.760    45.0      0.8694

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

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