g01 Chapter Contents
g01 Chapter Introduction
NAG C 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:     ltailIntegerInput
On entry: the length of the array tail.
Constraint: ${\mathbf{ltail}}>0$.
2:     tail[${\mathbf{ltail}}$]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:     lfIntegerInput
On entry: the length of the array f.
Constraint: ${\mathbf{lf}}>0$.
4:     f[lf]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:     ldf1IntegerInput
On entry: the length of the array df1.
Constraint: ${\mathbf{ldf1}}>0$.
6:     df1[ldf1]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:     ldf2IntegerInput
On entry: the length of the array df2.
Constraint: ${\mathbf{ldf2}}>0$.
8:     df2[ldf2]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:     p[$\mathit{dim}$]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:   ivalid[$\mathit{dim}$]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:   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, $\text{array size}=〈\mathit{\text{value}}〉$.
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.
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.

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)