hide long namesshow long names
hide short namesshow short names
Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

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

NAG Toolbox: nag_stat_inv_cdf_chisq_vector (g01tc)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_stat_inv_cdf_chisq_vector (g01tc) returns a number of deviates associated with the given probabilities of the χ2-distribution with real degrees of freedom.


[x, ivalid, ifail] = g01tc(tail, p, df, 'ltail', ltail, 'lp', lp, 'ldf', ldf)
[x, ivalid, ifail] = nag_stat_inv_cdf_chisq_vector(tail, p, df, 'ltail', ltail, 'lp', lp, 'ldf', ldf)


The deviate, xpi, associated with the lower tail probability pi of the χ2-distribution with νi degrees of freedom is defined as the solution to
P Xi xpi :νi = pi = 1 2 νi/2 Γ νi/2 0 xpi e -Xi/2 Xi vi / 2 - 1 dXi ,   0 xpi < ; ​ νi > 0 .  
The required xpi is found by using the relationship between a χ2-distribution and a gamma distribution, i.e., a χ2-distribution with νi degrees of freedom is equal to a gamma distribution with scale parameter 2 and shape parameter νi/2.
For very large values of νi, greater than 105, Wilson and Hilferty's Normal approximation to the χ2 is used; see Kendall and Stuart (1969).
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.


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:     tailltail – cell array of strings
Indicates which tail the supplied probabilities represent. For j= i-1 mod ltail +1 , for i=1,2,,maxltail,lp,ldf:
The lower tail probability, i.e., pi = P Xi xpi :νi .
The upper tail probability, i.e., pi = P Xi xpi :νi .
Constraint: tailj='L' or 'U', for j=1,2,,ltail.
2:     plp – double array
pi, the probability of the required χ2-distribution as defined by tail with pi=pj, j=i-1 mod lp+1.
  • if tailk='L', 0.0pj<1.0;
  • otherwise 0.0<pj1.0.
Where k=i-1 mod ltail+1 and j=i-1 mod lp+1.
3:     dfldf – double array
νi, the degrees of freedom of the χ2-distribution with νi=dfj, j=i-1 mod ldf+1.
Constraint: dfj>0.0, for j=1,2,,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:     lp int64int32nag_int scalar
Default: the dimension of the array p.
The length of the array p.
Constraint: lp>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:     x: – double array
The dimension of the array x will be maxltail,lp,ldf
xpi, the deviates for the χ2-distribution.
2:     ivalid: int64int32nag_int array
The dimension of the array ivalid will be maxltail,lp,ldf
ivalidi indicates any errors with the input arguments, with
No error.
On entry,invalid value supplied in tail when calculating xpi.
On entry,invalid value for pi.
On entry,νi0.0.
pi is too close to 0.0 or 1.0 for the result to be calculated.
The solution has failed to converge. The result should be a reasonable approximation.
3:     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 tail, p or df was invalid, or the solution failed to converge.
Check ivalid for more information.
Constraint: ltail>0.
Constraint: lp>0.
Constraint: ldf>0.
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 pi close to 0.0 or 1.0.

Further Comments

For higher accuracy the relationship described in Description may be used and a direct call to nag_stat_inv_cdf_gamma_vector (g01tf) made.


This example reads lower tail probabilities for several χ2-distributions, and calculates and prints the corresponding deviates.
function g01tc_example

fprintf('g01tc example results\n\n');

tail = {'L'};
p    = [0.01; 0.428; 0.869];
df   = [20;   7.5;  45];
[x, ivalid, ifail] = g01tc( ...
                            tail, p, df);

fprintf('  tail     p       df      x     ivalid\n');
ltail = numel(tail);
lp    = numel(p);
ldf   = numel(df);
len   = max ([ltail, lp, ldf]);
for i=0:len-1
  fprintf('%5s%8.3f%8.3f%8.3f%8d\n',  tail{mod(i, ltail)+1}, ...
          p(mod(i,lp)+1), df(mod(i,ldf)+1), x(i+1), ivalid(i+1));

g01tc example results

  tail     p       df      x     ivalid
    L   0.010  20.000   8.260       0
    L   0.428   7.500   6.201       0
    L   0.869  45.000  55.738       0

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

© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2015