NAG CL Interface
g01gdc (prob_​f_​noncentral)

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1 Purpose

g01gdc returns the probability associated with the lower tail of the noncentral F or variance-ratio distribution.

2 Specification

#include <nag.h>
double  g01gdc (double f, double df1, double df2, double lambda, double tol, Integer max_iter, NagError *fail)
The function may be called by the names: g01gdc, nag_stat_prob_f_noncentral or nag_prob_non_central_f_dist.

3 Description

The lower tail probability of the noncentral F-distribution with ν1 and ν2 degrees of freedom and noncentrality parameter λ, P(Ff:ν1,ν2;λ), is defined by
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+2j-2)/2[ν2+(ν1+2j)u] -(ν1+2j+ν2)/2  
and B(·,·) is the beta function.
The probability is computed by means of a transformation to a noncentral beta distribution:
where x= ν1f ν1f+ν2 and Pβ(Xx: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.

4 References

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

5 Arguments

1: f double Input
On entry: f, the deviate from the noncentral F-distribution.
Constraint: f>0.0.
2: df1 double Input
On entry: the degrees of freedom of the numerator variance, ν1.
Constraint: 0.0<df1106.
3: df2 double Input
On entry: the degrees of freedom of the denominator variance, ν2.
Constraint: df2>0.0.
4: lambda double Input
On entry: λ, the noncentrality parameter.
Constraint: 0.0lambda-2.0log(U) where U is the safe range parameter as defined by X02AMC.
5: tol double Input
On entry: the relative accuracy required by you in the results. If g01gdc is entered with tol greater than or equal to 1.0 or less than 10×machine precision (see X02AJC), the value of 10×machine precision is used instead.
6: max_iter Integer Input
On entry: the maximum number of iterations to be used.
Suggested value: 500. See g01gcc and g01gec for further details.
Constraint: max_iter1.
7: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

If on exit fail.code= NE_INT_ARG_LT, NE_PROB_F, NE_REAL_ARG_CONS or NE_REAL_ARG_LE, then g01gdc returns 0.0.
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
The solution has failed to converge in value iterations. Consider increasing max_iter or tol.
On entry, max_iter=value.
Constraint: max_iter1.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
The required probability cannot be computed accurately. This may happen if the result would be very close to zero or one. 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).
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 g01gdc returns zero otherwise the value returned should be an approximation to the correct value.
On entry, df1=value.
Constraint: 0.0<df1106.
On entry, df1=value.
Constraint: df1>0.0.
On entry, lambda=value.
Constraint: 0.0lambda-2.0×log(U), where U is the safe range parameter as defined by X02AMC.
On entry, df2=value.
Constraint: df2>0.0.
On entry, f=value.
Constraint: f>0.0.

7 Accuracy

The relative accuracy should be as specified by tol. For further details see g01gcc and g01gec.

8 Parallelism and Performance

g01gdc is not threaded in any implementation.

9 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.

10 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.

10.1 Program Text

Program Text (g01gdce.c)

10.2 Program Data

Program Data (g01gdce.d)

10.3 Program Results

Program Results (g01gdce.r)