nag_prob_f_vector (g01sdc) (PDF version)
g01 Chapter Contents
g01 Chapter Introduction
NAG C Library Manual

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:     ltailIntegerInput
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:     lfIntegerInput
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:     ldf1IntegerInput
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:     ldf2IntegerInput
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:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
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.
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  Further Comments

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

9  Example

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

9.1  Program Text

Program Text (g01sdce.c)

9.2  Program Data

Program Data (g01sdce.d)

9.3  Program Results

Program Results (g01sdce.r)


nag_prob_f_vector (g01sdc) (PDF version)
g01 Chapter Contents
g01 Chapter Introduction
NAG C Library Manual

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