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NAG Toolbox: nag_stat_prob_f_noncentral (g01gd)


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


[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:
Mark 23: tol now optional (default 0)


The lower tail probability of the noncentral FF-distribution with ν1ν1 and ν2ν2 degrees of freedom and noncentrality parameter λλ, P(Ff : ν1,ν2;λ)P(Ff:ν1,ν2;λ), is defined by
P(Ff : ν1,ν2;λ) = p(F : ν1,ν2;λ)dF,
P(F : ν1,ν2;λ ) = eλ / 2((λ / 2)j)/(j ! ) × ((ν1 + 2j)(ν1 + 2j) / 2 ν2ν2 / 2)/(B((ν1 + 2j) / 2,ν2 / 2))
j = 0
P(F : ν1,ν2;λ )=j= 0e-λ/2 (λ/2)jj! ×(ν1+ 2j)(ν1+ 2j)/2 ν2ν2/2 B((ν1+ 2j)/2,ν2/2)
× u(ν1 + 2j2) / 2[ν2 + (ν1 + 2j)u](ν1 + 2j + ν2) / 2
×u(ν1+2j-2)/2[ν2+(ν1+2j)u] -(ν1+2j+ν2)/2
and B( · , · )B(·,·) is the beta function.
The probability is computed by means of a transformation to a noncentral beta distribution:
P(Ff : ν1,ν2;λ) = Pβ(Xx : a,b;λ),
where x = (ν1f)/(ν1f + ν2) x= ν1f ν1f+ν2  and Pβ(Xx : a,b;λ)Pβ(Xx:a,b;λ) is the lower tail probability integral of the noncentral beta distribution with parameters aa, bb, and λλ.
If ν2ν2 is very large, greater than 106106, then a χ2χ2 approximation is used.


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


Compulsory Input Parameters

1:     f – double scalar
ff, the deviate from the noncentral FF-distribution.
Constraint: f > 0.0f>0.0.
2:     df1 – double scalar
The degrees of freedom of the numerator variance, ν1ν1.
Constraint: 0.0 < df11060.0<df1106.
3:     df2 – double scalar
The degrees of freedom of the denominator variance, ν2ν2.
Constraint: df2 > 0.0df2>0.0.
4:     rlamda – double scalar
λλ, the noncentrality parameter.
Constraint: 0.0rlamda2.0log(U)0.0rlamda-2.0log(U) where UU is the safe range parameter as defined by nag_machine_real_safe (x02am).

Optional Input Parameters

1:     tol – double scalar
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 1.01.0 or less than 10 × machine precision10×machine precision (see nag_machine_precision (x02aj)), then the value of 10 × machine precision10×machine precision is used instead.
Default: 0.00.0
2:     maxit – int64int32nag_int scalar
The maximum number of iterations to be used.
Constraint: maxit1maxit1.

Input Parameters Omitted from the MATLAB Interface


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_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 = 1ifail=1 or 33, then nag_stat_prob_f_noncentral (g01gd) 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,df10.0df10.0,
ordf1 > 106df1>106,
orrlamda < 0.0rlamda<0.0,
ormaxit < 1maxit<1,
orrlamda > 2.0log(U)rlamda>-2.0log(U), where U = U= safe range parameter as defined by nag_machine_real_safe (x02am).
  ifail = 2ifail=2
The solution has failed to converge in maxit iterations. You should try a larger value of maxit or tol.
  ifail = 3ifail=3
The required probability cannot be computed accurately. This may happen if the result would be very close to 0.00.0 or 1.01.0. 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 = 4ifail=4
The required accuracy was not achieved when calculating the initial value of the central FF (or χ2χ2) probability. You should try a larger value of tol. If the χ2χ2 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.


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 ν1ν1 and ν2ν2 are large a Normal approximation may be used and when only ν1ν1 is large a χ2χ2 approximation may be used. In both cases λλ is required to be of the same order as ν1ν1. See Abramowitz and Stegun (1972) for further details.


function nag_stat_prob_f_noncentral_example
f = 5.5;
df1 = 1.5;
df2 = 25.5;
rlamda = 3;
[result, ifail] = nag_stat_prob_f_noncentral(f, df1, df2, rlamda)

result =


ifail =


function g01gd_example
f = 5.5;
df1 = 1.5;
df2 = 25.5;
rlamda = 3;
[result, ifail] = g01gd(f, df1, df2, rlamda)

result =


ifail =


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

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