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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_stat_prob_chisq_vector (g01sc)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_stat_prob_chisq_vector (g01sc) returns a number of lower or upper tail probabilities for the χ2-distribution with real degrees of freedom.


[p, ivalid, ifail] = g01sc(tail, x, df, 'ltail', ltail, 'lx', lx, 'ldf', ldf)
[p, ivalid, ifail] = nag_stat_prob_chisq_vector(tail, x, df, 'ltail', ltail, 'lx', lx, 'ldf', ldf)


The lower tail probability for the χ2-distribution with νi degrees of freedom, P = Xi xi :νi  is defined by:
P = Xixi:νi = 1 2 νi/2 Γ νi/2 0.0 xi Xi νi/2-1 e -Xi/2 dXi ,   xi 0 , νi > 0 .  
To calculate P = Xi xi :νi  a transformation of a gamma distribution is employed, 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.
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.


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:     tailltail – 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,,maxltail,lx,ldf:
The lower tail probability is returned, i.e., pi = P Xi xi :νi .
The upper tail probability is returned, i.e., pi = P Xi xi :νi .
Constraint: tailj='L' or 'U', for j=1,2,,ltail.
2:     xlx – double array
xi, the values of the χ2 variates with νi degrees of freedom with xi=xj, j=i-1 mod lx+1.
Constraint: xj0.0, for j=1,2,,lx.
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:     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 maxltail,ldf,lx
pi, the probabilities for the χ2 distribution.
2:     ivalid: int64int32nag_int array
The dimension of the array ivalid will be maxltail,ldf,lx
ivalidi indicates any errors with the input arguments, with
No error.
On entry,invalid value supplied in tail when calculating pi.
On entry,xi<0.0.
On entry,νi0.0.
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.
Constraint: ltail>0.
Constraint: lx>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.


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.
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));

g01sc example results

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

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Chapter Contents
Chapter Introduction
NAG Toolbox

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