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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_stat_inv_cdf_chisq (g01fc)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_stat_inv_cdf_chisq (g01fc) returns the deviate associated with the given lower tail probability of the χ2-distribution with real degrees of freedom.


[result, ifail] = g01fc(p, df)
[result, ifail] = nag_stat_inv_cdf_chisq(p, df)


The deviate, xp, associated with the lower tail probability p of the χ2-distribution with ν degrees of freedom is defined as the solution to
PXxp:ν=p=12ν/2Γν/2 0xpe-X/2Xv/2-1dX,  0xp<;ν>0.  
The required xp is found by using the relationship between a χ2-distribution and a gamma distribution, 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.
For very large values of ν, greater than 105, Wilson and Hilferty's normal approximation to the χ2 is used; see Kendall and Stuart (1969).


Best D J and Roberts D E (1975) Algorithm AS 91. The percentage points of the χ2 distribution Appl. Statist. 24 385–388
Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth
Kendall M G and Stuart A (1969) The Advanced Theory of Statistics (Volume 1) (3rd Edition) Griffin


Compulsory Input Parameters

1:     p – double scalar
p, the lower tail probability from the required χ2-distribution.
Constraint: 0.0p<1.0.
2:     df – double scalar
ν, the degrees of freedom of the χ2-distribution.
Constraint: df>0.0.

Optional Input Parameters


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_inv_cdf_chisq (g01fc) 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, 3 or 5 on exit, then nag_stat_inv_cdf_chisq (g01fc) returns 0.0.

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

On entry,p<0.0,
On entry,df0.0.
p is too close to 0 or 1 for the result to be calculated.
W  ifail=4
The solution has failed to converge. The result should be a reasonable approximation.
The series used to calculate the gamma function has failed to converge. This is an unlikely error exit.
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.


The results should be accurate to five significant digits for most argument values. Some accuracy is lost for p close to 0.0.

Further Comments

For higher accuracy the relationship described in Description may be used and a direct call to nag_stat_inv_cdf_gamma (g01ff) made.


This example reads lower tail probabilities for several χ2-distributions, and calculates and prints the corresponding deviates until the end of data is reached.
function g01fc_example

fprintf('g01fc example results\n\n');

p    = [ 0.01;   0.428;   0.869];
df   = [20.00;   7.500;  45.000];

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

  [x, ifail] = g01fc( ...
                      p(j) , df(j));

  fprintf('%9.3f%8.3f%8.3f\n', p(j), df(j), x);

g01fc example results

      p      df       x
    0.010  20.000   8.260
    0.428   7.500   6.201
    0.869  45.000  55.738

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

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