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NAG Toolbox

NAG Toolbox: nag_stat_inv_cdf_chisq (g01fc)

Purpose

nag_stat_inv_cdf_chisq (g01fc) returns the deviate associated with the given lower tail probability of the χ2χ2-distribution with real degrees of freedom.

Syntax

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

Description

The deviate, xpxp, associated with the lower tail probability pp of the χ2χ2-distribution with νν degrees of freedom is defined as the solution to
xp
P(Xxp : ν) = p = 1/(2ν / 2Γ(ν / 2))eX / 2Xv / 21dX,  0xp < ;ν > 0.
0
P(Xxp:ν)=p=12ν/2Γ(ν/2) 0xpe-X/2Xv/2-1dX,  0xp<;ν>0.
The required xpxp is found by using the relationship between a χ2χ2-distribution and a gamma distribution, i.e., a χ2χ2-distribution with νν degrees of freedom is equal to a gamma distribution with scale parameter 22 and shape parameter ν / 2ν/2.
For very large values of νν, greater than 105105, Wilson and Hilferty's normal approximation to the χ2χ2 is used; see Kendall and Stuart (1969).

References

Best D J and Roberts D E (1975) Algorithm AS 91. The percentage points of the χ2χ2 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

Parameters

Compulsory Input Parameters

1:     p – double scalar
pp, the lower tail probability from the required χ2χ2-distribution.
Constraint: 0.0p < 1.00.0p<1.0.
2:     df – double scalar
νν, the degrees of freedom of the χ2χ2-distribution.
Constraint: df > 0.0df>0.0.

Optional Input Parameters

None.

Input Parameters Omitted from the MATLAB Interface

None.

Output Parameters

1:     result – double scalar
The result of the function.
2:     ifail – int64int32nag_int scalar
ifail = 0ifail=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 = 1ifail=1, 22, 33 or 55 on exit, then nag_stat_inv_cdf_chisq (g01fc) returns 0.00.0.

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

  ifail = 1ifail=1
On entry,p < 0.0p<0.0,
orp1.0p1.0.
  ifail = 2ifail=2
On entry,df0.0df0.0.
  ifail = 3ifail=3
p is too close to 00 or 11 for the result to be calculated.
W ifail = 4ifail=4
The solution has failed to converge. The result should be a reasonable approximation.
  ifail = 5ifail=5
The series used to calculate the gamma function has failed to converge. This is an unlikely error exit.

Accuracy

The results should be accurate to five significant digits for most parameter values. Some accuracy is lost for pp close to 0.00.0.

Further Comments

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

Example

function nag_stat_inv_cdf_chisq_example
p = 0.01;
df = 20;
[result, ifail] = nag_stat_inv_cdf_chisq(p, df)
 

result =

    8.2604


ifail =

                    0


function g01fc_example
p = 0.01;
df = 20;
[result, ifail] = g01fc(p, df)
 

result =

    8.2604


ifail =

                    0



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

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