NAG Library Function Document

nag_prob_f_vector (g01sdc)

 Contents

    1  Purpose
    7  Accuracy

1
Purpose

nag_prob_f_vector (g01sdc) returns a number of lower or upper tail probabilities for the F or variance-ratio distribution with real degrees of freedom.

2
Specification

#include <nag.h>
#include <nagg01.h>
void  nag_prob_f_vector (Integer ltail, const Nag_TailProbability tail[], Integer lf, const double f[], Integer ldf1, const double df1[], Integer ldf2, const double df2[], double p[], Integer ivalid[], NagError *fail)

3
Description

The lower tail probability for the F, or variance-ratio, distribution with ui and vi degrees of freedom, P Fi fi :ui,vi , is defined by:
P Fi fi :ui,vi = ui ui/2 vi vi/2 Γ ui + vi / 2 Γ ui/2 Γ vi/2 0 fi Fi ui-2 / 2 ui Fi + vi - ui + vi / 2 d Fi ,  
for ui, vi>0, fi0.
The probability is computed by means of a transformation to a beta distribution, Pβi Bi βi :ai,bi :
P Fi fi :ui,vi = Pβi Bi ui fi ui fi + vi : ui / 2 , vi / 2  
and using a call to nag_prob_beta_dist (g01eec).
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 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
Arguments

1:     ltail IntegerInput
On entry: the length of the array tail.
Constraint: ltail>0.
2:     tail[ltail] const Nag_TailProbabilityInput
On entry: indicates whether the lower or upper tail probabilities are required. For j= i-1 mod ltail , for i=1,2,,maxltail,lf,ldf1,ldf2:
tail[j]=Nag_LowerTail
The lower tail probability is returned, i.e., pi = P Fi fi :ui,vi .
tail[j]=Nag_UpperTail
The upper tail probability is returned, i.e., pi = P Fi fi :ui,vi .
Constraint: tail[j-1]=Nag_LowerTail or Nag_UpperTail, for j=1,2,,ltail.
3:     lf IntegerInput
On entry: the length of the array f.
Constraint: lf>0.
4:     f[lf] const doubleInput
On entry: fi, the value of the F variate with fi=f[j], j=i-1 mod lf.
Constraint: f[j-1]0.0, for j=1,2,,lf.
5:     ldf1 IntegerInput
On entry: the length of the array df1.
Constraint: ldf1>0.
6:     df1[ldf1] const doubleInput
On entry: ui, the degrees of freedom of the numerator variance with ui=df1[j], j=i-1 mod ldf1.
Constraint: df1[j-1]>0.0, for j=1,2,,ldf1.
7:     ldf2 IntegerInput
On entry: the length of the array df2.
Constraint: ldf2>0.
8:     df2[ldf2] const doubleInput
On entry: vi, the degrees of freedom of the denominator variance with vi=df2[j], j=i-1 mod ldf2.
Constraint: df2[j-1]>0.0, for j=1,2,,ldf2.
9:     p[dim] doubleOutput
Note: the dimension, dim, of the array p must be at least maxltail,lf,ldf1,ldf2.
On exit: pi, the probabilities for the F-distribution.
10:   ivalid[dim] IntegerOutput
Note: the dimension, dim, of the array ivalid must be at least maxltail,lf,ldf1,ldf2.
On exit: ivalid[i-1] indicates any errors with the input arguments, with
ivalid[i-1]=0
No error.
ivalid[i-1]=1
On entry,invalid value supplied in tail when calculating pi.
ivalid[i-1]=2
On entry,fi<0.0.
ivalid[i-1]=3
On entry,ui0.0,
orvi0.0.
ivalid[i-1]=4
The solution has failed to converge. The result returned should represent an approximation to the solution.
11:   fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6
Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_ARRAY_SIZE
On entry, array size=value.
Constraint: ldf1>0.
On entry, array size=value.
Constraint: ldf2>0.
On entry, array size=value.
Constraint: lf>0.
On entry, array size=value.
Constraint: ltail>0.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INTERNAL_ERROR
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NW_IVALID
On entry, at least one value of f, df1, df2 or tail was invalid, or the solution failed to converge.
Check ivalid for more information.

7
Accuracy

The result should be accurate to five significant digits.

8
Parallelism and Performance

nag_prob_f_vector (g01sdc) is not threaded in any implementation.

9
Further Comments

For higher accuracy nag_prob_beta_vector (g01sec) can be used along with the transformations given in Section 3.

10
Example

This example reads values from, and degrees of freedom for, a number of F-distributions and computes the associated lower tail probabilities.

10.1
Program Text

Program Text (g01sdce.c)

10.2
Program Data

Program Data (g01sdce.d)

10.3
Program Results

Program Results (g01sdce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017