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NAG Toolbox: nag_stat_prob_f_noncentral (g01gd)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_stat_prob_f_noncentral (g01gd) returns the probability associated with the lower tail of the noncentral F or variance-ratio distribution.

Syntax

[result, ifail] = g01gd(f, df1, df2, rlamda, 'tol', tol, 'maxit', maxit)
[result, ifail] = nag_stat_prob_f_noncentral(f, df1, df2, rlamda, 'tol', tol, 'maxit', maxit)
Note: the interface to this routine has changed since earlier releases of the toolbox:
At Mark 23: tol was made optional (default 0)

Description

The lower tail probability of the noncentral F-distribution with ν1 and ν2 degrees of freedom and noncentrality parameter λ, PFf:ν1,ν2;λ, is defined by
PFf:ν1,ν2;λ=0xpF:ν1,ν2;λdF,  
where
PF : ν1,ν2;λ =j= 0e-λ/2 λ/2jj! ×ν1+ 2jν1+ 2j/2 ν2ν2/2 Bν1+ 2j/2,ν2/2  
×uν1+2j-2/2ν2+ν1+2ju -ν1+2j+ν2/2  
and B·,· is the beta function.
The probability is computed by means of a transformation to a noncentral beta distribution:
PFf:ν1,ν2;λ=PβXx:a,b;λ,  
where x= ν1f ν1f+ν2  and PβXx:a,b;λ is the lower tail probability integral of the noncentral beta distribution with parameters a, b, and λ.
If ν2 is very large, greater than 106, then a χ2 approximation is used.

References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

Parameters

Compulsory Input Parameters

1:     f – double scalar
f, the deviate from the noncentral F-distribution.
Constraint: f>0.0.
2:     df1 – double scalar
The degrees of freedom of the numerator variance, ν1.
Constraint: 0.0<df1106.
3:     df2 – double scalar
The degrees of freedom of the denominator variance, ν2.
Constraint: df2>0.0.
4:     rlamda – double scalar
λ, the noncentrality parameter.
Constraint: 0.0rlamda-2.0logU where U is the safe range parameter as defined by nag_machine_real_safe (x02am).

Optional Input Parameters

1:     tol – double scalar
Default: 0.0
The relative accuracy required by you in the results. If nag_stat_prob_f_noncentral (g01gd) is entered with tol greater than or equal to 1.0 or less than 10×machine precision (see nag_machine_precision (x02aj)), then the value of 10×machine precision is used instead.
2:     maxit int64int32nag_int scalar
The maximum number of iterations to be used.
Constraint: maxit1.

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_prob_f_noncentral (g01gd) may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the function:
If on exit ifail=1 or 3, then nag_stat_prob_f_noncentral (g01gd) 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.

   ifail=1
On entry,df10.0,
ordf1>106,
ordf20.0,
orf0.0,
orrlamda<0.0,
ormaxit<1,
orrlamda>-2.0logU, where U= safe range argument as defined by nag_machine_real_safe (x02am).
   ifail=2
The solution has failed to converge in maxit iterations. You should try a larger value of maxit or tol.
   ifail=3
The required probability cannot be computed accurately. This may happen if the result would be very close to 0.0 or 1.0. Alternatively the values of df1 and f may be too large. In the latter case you could try using a normal approximation; see Abramowitz and Stegun (1972).
W  ifail=4
The required accuracy was not achieved when calculating the initial value of the central F (or χ2) probability. You should try a larger value of tol. If the χ2 approximation is being used then nag_stat_prob_f_noncentral (g01gd) returns zero otherwise the value returned should be an approximation to the correct value.
   ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
   ifail=-399
Your licence key may have expired or may not have been installed correctly.
   ifail=-999
Dynamic memory allocation failed.

Accuracy

The relative accuracy should be as specified by tol. For further details see nag_stat_prob_chisq_noncentral (g01gc) and nag_stat_prob_beta_noncentral (g01ge).

Further Comments

When both ν1 and ν2 are large a Normal approximation may be used and when only ν1 is large a χ2 approximation may be used. In both cases λ is required to be of the same order as ν1. See Abramowitz and Stegun (1972) for further details.

Example

This example reads values from, and degrees of freedom for, F-distributions, computes the lower tail probabilities and prints all these values until the end of data is reached.
function g01gd_example


fprintf('g01gd example results\n\n');

f      = [  5.5     39.9     2.5 ];
df1    = [  1.5      1      20.25];
df2    = [ 25.5      1       1   ];
rlamda = [  3        2       0   ];
p      = f;

fprintf('     f      df1     df2    rlamda     p\n');
for j = 1:numel(f)
   [p(j), ifail] = g01gd( ...
			  f(j), df1(j), df2(j), rlamda(j));
end

fprintf('%8.3f%8.3f%8.3f%8.3f%8.4f\n', [f; df1; df2; rlamda; p]);


g01gd example results

     f      df1     df2    rlamda     p
   5.500   1.500  25.500   3.000  0.8214
  39.900   1.000   1.000   2.000  0.8160
   2.500  20.250   1.000   0.000  0.5342

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Chapter Introduction
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