g01 Chapter Contents
g01 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_prob_f_vector (g01sdc)

## 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 #include
 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 ${u}_{i}$ and ${v}_{i}$ degrees of freedom, $P\left({F}_{i}\le {f}_{i}:{u}_{i},{v}_{i}\right)$, 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 ${u}_{i}$, ${v}_{i}>0$, ${f}_{i}\ge 0$.
The probability is computed by means of a transformation to a beta distribution, ${P}_{{\beta }_{i}}\left({B}_{i}\le {\beta }_{i}:{a}_{i},{b}_{i}\right)$:
 $P Fi ≤ fi :ui,vi = Pβi Bi ≤ ui fi ui fi + vi : ui / 2 , vi / 2$
and using a call to nag_prob_beta_dist (g01eec).
For very large values of both ${u}_{i}$ and ${v}_{i}$, greater than ${10}^{5}$, a normal approximation is used. If only one of ${u}_{i}$ or ${v}_{i}$ is greater than ${10}^{5}$ then a ${\chi }^{2}$ 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:    $\mathbf{ltail}$IntegerInput
On entry: the length of the array tail.
Constraint: ${\mathbf{ltail}}>0$.
2:    $\mathbf{tail}\left[{\mathbf{ltail}}\right]$const Nag_TailProbabilityInput
On entry: indicates whether the lower or upper tail probabilities are required. For , for $\mathit{i}=1,2,\dots ,\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lf}},{\mathbf{ldf1}},{\mathbf{ldf2}}\right)$:
${\mathbf{tail}}\left[j\right]=\mathrm{Nag_LowerTail}$
The lower tail probability is returned, i.e., ${p}_{i}=P\left({F}_{i}\le {f}_{i}:{u}_{i},{v}_{i}\right)$.
${\mathbf{tail}}\left[j\right]=\mathrm{Nag_UpperTail}$
The upper tail probability is returned, i.e., ${p}_{i}=P\left({F}_{i}\ge {f}_{i}:{u}_{i},{v}_{i}\right)$.
Constraint: ${\mathbf{tail}}\left[\mathit{j}-1\right]=\mathrm{Nag_LowerTail}$ or $\mathrm{Nag_UpperTail}$, for $\mathit{j}=1,2,\dots ,{\mathbf{ltail}}$.
3:    $\mathbf{lf}$IntegerInput
On entry: the length of the array f.
Constraint: ${\mathbf{lf}}>0$.
4:    $\mathbf{f}\left[{\mathbf{lf}}\right]$const doubleInput
On entry: ${f}_{i}$, the value of the $F$ variate with ${f}_{i}={\mathbf{f}}\left[j\right]$, .
Constraint: ${\mathbf{f}}\left[\mathit{j}-1\right]\ge 0.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{lf}}$.
5:    $\mathbf{ldf1}$IntegerInput
On entry: the length of the array df1.
Constraint: ${\mathbf{ldf1}}>0$.
6:    $\mathbf{df1}\left[{\mathbf{ldf1}}\right]$const doubleInput
On entry: ${u}_{i}$, the degrees of freedom of the numerator variance with ${u}_{i}={\mathbf{df1}}\left[j\right]$, .
Constraint: ${\mathbf{df1}}\left[\mathit{j}-1\right]>0.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{ldf1}}$.
7:    $\mathbf{ldf2}$IntegerInput
On entry: the length of the array df2.
Constraint: ${\mathbf{ldf2}}>0$.
8:    $\mathbf{df2}\left[{\mathbf{ldf2}}\right]$const doubleInput
On entry: ${v}_{i}$, the degrees of freedom of the denominator variance with ${v}_{i}={\mathbf{df2}}\left[j\right]$, .
Constraint: ${\mathbf{df2}}\left[\mathit{j}-1\right]>0.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{ldf2}}$.
9:    $\mathbf{p}\left[\mathit{dim}\right]$doubleOutput
Note: the dimension, dim, of the array p must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lf}},{\mathbf{ldf1}},{\mathbf{ldf2}}\right)$.
On exit: ${p}_{i}$, the probabilities for the $F$-distribution.
10:  $\mathbf{ivalid}\left[\mathit{dim}\right]$IntegerOutput
Note: the dimension, dim, of the array ivalid must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lf}},{\mathbf{ldf1}},{\mathbf{ldf2}}\right)$.
On exit: ${\mathbf{ivalid}}\left[i-1\right]$ indicates any errors with the input arguments, with
${\mathbf{ivalid}}\left[i-1\right]=0$
No error.
${\mathbf{ivalid}}\left[i-1\right]=1$
 On entry, invalid value supplied in tail when calculating ${p}_{i}$.
${\mathbf{ivalid}}\left[i-1\right]=2$
 On entry, ${f}_{i}<0.0$.
${\mathbf{ivalid}}\left[i-1\right]=3$
 On entry, ${u}_{i}\le 0.0$, or ${v}_{i}\le 0.0$.
${\mathbf{ivalid}}\left[i-1\right]=4$
The solution has failed to converge. The result returned should represent an approximation to the solution.
11:  $\mathbf{fail}$NagError *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.
See Section 3.2.1.2 in the Essential Introduction for further information.
NE_ARRAY_SIZE
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldf1}}>0$.
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldf2}}>0$.
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lf}}>0$.
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ltail}}>0$.
On entry, argument $〈\mathit{\text{value}}〉$ 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 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction 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.

## 7  Accuracy

The result should be accurate to five significant digits.

## 8  Parallelism and Performance

Not applicable.

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)