G01TDF (PDF version)
G01 Chapter Contents
G01 Chapter Introduction
NAG Library Manual

NAG Library Routine Document


Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

G01TDF returns a number of deviates associated with given probabilities of the F or variance-ratio distribution with real degrees of freedom.

2  Specification

REAL (KIND=nag_wp)  P(LP), DF1(LDF1), DF2(LDF2), F(*)

3  Description

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 G01TEF.
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 routine are designed to allow maximum flexibility in the supply of vector parameters by re-using elements of any arrays that are shorter than the total number of evaluations required. See Section 2.6 in the G01 Chapter Introduction for further information.

4  References

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

5  Parameters

1:     LTAIL – INTEGERInput
On entry: the length of the array TAIL.
Constraint: LTAIL>0.
2:     TAIL(LTAIL) – CHARACTER(1) arrayInput
On entry: 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.
3:     LP – INTEGERInput
On entry: the length of the array P.
Constraint: LP>0.
4:     P(LP) – REAL (KIND=nag_wp) arrayInput
On entry: 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.
5:     LDF1 – INTEGERInput
On entry: the length of the array DF1.
Constraint: LDF1>0.
6:     DF1(LDF1) – REAL (KIND=nag_wp) arrayInput
On entry: 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.
7:     LDF2 – INTEGERInput
On entry: the length of the array DF2.
Constraint: LDF2>0.
8:     DF2(LDF2) – REAL (KIND=nag_wp) arrayInput
On entry: 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.
9:     F(*) – REAL (KIND=nag_wp) arrayOutput
Note: the dimension of the array F must be at least maxLTAIL,LP,LDF1,LDF2.
On exit: fpi, the deviates for the F-distribution.
10:   IVALID(*) – INTEGER arrayOutput
Note: the dimension of the array IVALID must be at least maxLTAIL,LP,LDF1,LDF2.
On exit: 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.
11:   IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to 0, -1​ or ​1. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1​ or ​1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, because for this routine the values of the output parameters may be useful even if IFAIL0 on exit, the recommended value is -1. When the value -1​ or ​1 is used it is essential to test the value of IFAIL on exit.
On exit: IFAIL=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6  Error Indicators and Warnings

If on entry IFAIL=0 or -1, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Note: G01TDF may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the routine:
On entry, at least one value of TAIL, P, DF1, DF2 was invalid, or the solution failed to converge.
Check IVALID for more information.
On entry, array size=value.
Constraint: LTAIL>0.
On entry, array size=value.
Constraint: LP>0.
On entry, array size=value.
Constraint: LDF1>0.
On entry, array size=value.
Constraint: LDF2>0.
Dynamic memory allocation failed.

7  Accuracy

The result should be accurate to five significant digits.

8  Further Comments

For higher accuracy G01TEF can be used along with the transformations given in Section 3.

9  Example

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

9.1  Program Text

Program Text (g01tdfe.f90)

9.2  Program Data

Program Data (g01tdfe.d)

9.3  Program Results

Program Results (g01tdfe.r)

G01TDF (PDF version)
G01 Chapter Contents
G01 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012