# NAG Library Function Document

## 1Purpose

nag_deviates_f_vector (g01tdc) returns a number of deviates associated with given probabilities of the $F$ or variance-ratio distribution with real degrees of freedom.

## 2Specification

 #include #include
 void nag_deviates_f_vector (Integer ltail, const Nag_TailProbability tail[], Integer lp, const double p[], Integer ldf1, const double df1[], Integer ldf2, const double df2[], double f[], Integer ivalid[], NagError *fail)

## 3Description

The deviate, ${f}_{{p}_{i}}$, associated with the lower tail probability, ${p}_{i}$, of the $F$-distribution with degrees of freedom ${u}_{i}$ and ${v}_{i}$ 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 ${u}_{i},{v}_{i}>0$; $0\le {f}_{{p}_{i}}<\infty$.
The value of ${f}_{{p}_{i}}$ is computed by means of a transformation to a beta distribution, ${P}_{i{\beta }_{i}}\left({B}_{i}\le {\beta }_{i}:{a}_{i},{b}_{i}\right)$:
 $P Fi ≤ fpi :ui,vi = P iβi Bi ≤ ui fpi ui fpi + vi : ui / 2 , vi / 2$
and using a call to nag_deviates_beta_vector (g01tec).
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.

## 4References

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

## 5Arguments

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 which tail the supplied probabilities represent. For , for $\mathit{i}=1,2,\dots ,\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lp}},{\mathbf{ldf1}},{\mathbf{ldf2}}\right)$:
${\mathbf{tail}}\left[j\right]=\mathrm{Nag_LowerTail}$
The lower tail probability, i.e., ${p}_{i}=P\left({F}_{i}\le {f}_{{p}_{i}}:{u}_{i},{v}_{i}\right)$.
${\mathbf{tail}}\left[j\right]=\mathrm{Nag_UpperTail}$
The upper tail probability, i.e., ${p}_{i}=P\left({F}_{i}\ge {f}_{{p}_{i}}:{u}_{i},{v}_{i}\right)$.
Constraint: ${\mathbf{tail}}\left[\mathit{j}-1\right]=\mathrm{Nag_LowerTail}$ or $\mathrm{Nag_UpperTail}$, for $\mathit{j}=1,2,\dots ,{\mathbf{ltail}}$.
3:    $\mathbf{lp}$IntegerInput
On entry: the length of the array p.
Constraint: ${\mathbf{lp}}>0$.
4:    $\mathbf{p}\left[{\mathbf{lp}}\right]$const doubleInput
On entry: ${p}_{i}$, the probability of the required $F$-distribution as defined by tail with ${p}_{i}={\mathbf{p}}\left[j\right]$, .
Constraints:
• if ${\mathbf{tail}}\left[k\right]=\mathrm{Nag_LowerTail}$, $0.0\le {\mathbf{p}}\left[\mathit{j}\right]<1.0$;
• otherwise $0.0<{\mathbf{p}}\left[\mathit{j}\right]\le 1.0$.
Where  and .
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:    $\mathbf{f}\left[\mathit{dim}\right]$doubleOutput
Note: the dimension, dim, of the array f must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lp}},{\mathbf{ldf1}},{\mathbf{ldf2}}\right)$.
On exit: ${f}_{{p}_{i}}$, the deviates for the $F$-distribution.
10:  $\mathbf{ivalid}\left[\mathit{dim}\right]$IntegerOutput
Note: the dimension, dim, of the array ivalid must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lp}},{\mathbf{ldf1}},{\mathbf{ldf2}}\right)$.
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 ${f}_{{p}_{i}}$.
${\mathbf{ivalid}}\left[i-1\right]=2$
 On entry, invalid value for ${p}_{i}$.
${\mathbf{ivalid}}\left[i-1\right]=3$
 On entry, ${u}_{i}\le 0.0$, or ${v}_{i}\le 0.0$.
${\mathbf{ivalid}}\left[i-1\right]=4$
The solution has not converged. The result should still be a reasonable approximation to the solution.
${\mathbf{ivalid}}\left[i-1\right]=5$
The value of ${p}_{i}$ 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:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error 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, $\text{array size}=〈\mathit{\text{value}}〉$.
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: ${\mathbf{lp}}>0$.
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 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 tail, p, df1, df2 was invalid, or the solution failed to converge.

## 7Accuracy

The result should be accurate to five significant digits.

## 8Parallelism and Performance

nag_deviates_f_vector (g01tdc) is not threaded in any implementation.

For higher accuracy nag_deviates_beta_vector (g01tec) can be used along with the transformations given in Section 3.

## 10Example

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

### 10.1Program Text

Program Text (g01tdce.c)

### 10.2Program Data

Program Data (g01tdce.d)

### 10.3Program Results

Program Results (g01tdce.r)

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