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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_stat_inv_cdf_f_vector (g01td)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_stat_inv_cdf_f_vector (g01td) returns a number of deviates associated with given probabilities of the F or variance-ratio distribution with real degrees of freedom.


[f, ivalid, ifail] = g01td(tail, p, df1, df2, 'ltail', ltail, 'lp', lp, 'ldf1', ldf1, 'ldf2', ldf2)
[f, ivalid, ifail] = nag_stat_inv_cdf_f_vector(tail, p, df1, df2, 'ltail', ltail, 'lp', lp, 'ldf1', ldf1, 'ldf2', ldf2)


The deviate, fpi, associated with the lower tail probability, pi, of the F-distribution with degrees of freedom ui and vi is defined as the solution to
P Fi fpi :ui,vi = pi = u i 12 ui v i 12 vi Γ ui + vi 2 Γ ui 2 Γ vi 2 0 fpi Fi 12 ui-2 vi + ui Fi -12 ui + vi dFi ,  
where ui,vi>0; 0fpi<.
The value of fpi is computed by means of a transformation to a beta distribution, P iβi Bi βi :ai,bi :
P Fi fpi :ui,vi = P iβi Bi ui fpi ui fpi + vi : ui / 2 , vi / 2  
and using a call to nag_stat_inv_cdf_beta_vector (g01te).
For very large values of both ui and vi, greater than 105, a Normal approximation is used. If only one of ui or vi is greater than 105 then a χ2 approximation is used; see Abramowitz and Stegun (1972).
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 which tail the supplied probabilities represent. For j= i-1 mod ltail +1 , for i=1,2,,maxltail,lp,ldf1,ldf2:
The lower tail probability, i.e., pi = P Fi fpi : ui , vi .
The upper tail probability, i.e., pi = P Fi fpi : ui , vi .
Constraint: tailj='L' or 'U', for j=1,2,,ltail.
2:     plp – double array
pi, the probability of the required F-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:     df1ldf1 – double array
ui, the degrees of freedom of the numerator variance with ui=df1j, j=i-1 mod ldf1+1.
Constraint: df1j>0.0, for j=1,2,,ldf1.
4:     df2ldf2 – double array
vi, the degrees of freedom of the denominator variance with vi=df2j, j=i-1 mod ldf2+1.
Constraint: df2j>0.0, for j=1,2,,ldf2.

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:     ldf1 int64int32nag_int scalar
Default: the dimension of the array df1.
The length of the array df1.
Constraint: ldf1>0.
4:     ldf2 int64int32nag_int scalar
Default: the dimension of the array df2.
The length of the array df2.
Constraint: ldf2>0.

Output Parameters

1:     f: – double array
The dimension of the array f will be maxltail,lp,ldf1,ldf2
fpi, the deviates for the F-distribution.
2:     ivalid: int64int32nag_int array
The dimension of the array ivalid will be maxltail,lp,ldf1,ldf2
ivalidi indicates any errors with the input arguments, with
No error.
On entry,invalid value supplied in tail when calculating fpi.
On entry,invalid value for pi.
On entry,ui0.0,
The solution has not converged. The result should still be a reasonable approximation to the solution.
The value of pi is too close to 0.0 or 1.0 for the result to be computed. This will only occur when the large sample approximations are used.
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_inv_cdf_f_vector (g01td) 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 tail, p, df1, df2 was invalid, or the solution failed to converge.
Check ivalid for more information.
Constraint: ltail>0.
Constraint: lp>0.
Constraint: ldf1>0.
Constraint: ldf2>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 result should be accurate to five significant digits.

Further Comments

For higher accuracy nag_stat_inv_cdf_beta_vector (g01te) can be used along with the transformations given in Description.


This example reads the lower tail probabilities for several F-distributions, and calculates and prints the corresponding deviates.
function g01td_example

fprintf('g01td example results\n\n');

tail = {'L'};
p    = [0.984; 0.9; 0.534];
df1  = [10;    1;  20.25];
df2  = [25.5;  1;   1];

[f, ivalid, ifail] = g01td( ...
                            tail, p, df1, df2);

fprintf('   tail   p      df1     df2      f    ivalid\n');
ltail = numel(tail);
lp    = numel(p);
ldf1  = numel(df1);
ldf2  = numel(df2);
len   = max ([ltail, lp, ldf1, ldf2]);
for i=0:len-1
  fprintf('%5s%7.3f%8.3f%8.3f%8.3f%7d\n', tail{mod(i, ltail)+1}, ...
          p(mod(i,lp)+1), df1(mod(i,ldf1)+1), df2(mod(i,ldf2)+1), ...
          f(i+1), ivalid(i+1));

g01td example results

   tail   p      df1     df2      f    ivalid
    L  0.984  10.000  25.500   2.847      0
    L  0.900   1.000   1.000  39.863      0
    L  0.534  20.250   1.000   2.498      0

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

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