G01 Chapter Contents
G01 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentG01SDF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

G01SDF returns a number of lower or upper tail probabilities for the $F$ or variance-ratio distribution with real degrees of freedom.

## 2  Specification

 SUBROUTINE G01SDF ( LTAIL, TAIL, LF, F, LDF1, DF1, LDF2, DF2, P, IVALID, IFAIL)
 INTEGER LTAIL, LF, LDF1, LDF2, IVALID(*), IFAIL REAL (KIND=nag_wp) F(LF), DF1(LDF1), DF2(LDF2), P(*) CHARACTER(1) TAIL(LTAIL)

## 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 G01EEF.
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 routine are designed to allow maximum flexibility in the supply of vector parameters 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  Parameters

1:     LTAIL – INTEGERInput
On entry: the length of the array TAIL.
Constraint: ${\mathbf{LTAIL}}>0$.
2:     TAIL(LTAIL) – CHARACTER(1) arrayInput
On entry: indicates whether the lower or upper tail probabilities are required. For , for $\mathit{i}=1,2,\dots ,\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{LTAIL}},{\mathbf{LF}},{\mathbf{LDF1}},{\mathbf{LDF2}}\right)$:
${\mathbf{TAIL}}\left(j\right)=\text{'L'}$
The lower tail probability is returned, i.e., ${p}_{i}=P\left({F}_{i}\le {f}_{i}:{u}_{i},{v}_{i}\right)$.
${\mathbf{TAIL}}\left(j\right)=\text{'U'}$
The upper tail probability is returned, i.e., ${p}_{i}=P\left({F}_{i}\ge {f}_{i}:{u}_{i},{v}_{i}\right)$.
Constraint: ${\mathbf{TAIL}}\left(\mathit{j}\right)=\text{'L'}$ or $\text{'U'}$, for $\mathit{j}=1,2,\dots ,{\mathbf{LTAIL}}$.
3:     LF – INTEGERInput
On entry: the length of the array F.
Constraint: ${\mathbf{LF}}>0$.
4:     F(LF) – REAL (KIND=nag_wp) arrayInput
On entry: ${f}_{i}$, the value of the $F$ variate with ${f}_{i}={\mathbf{F}}\left(j\right)$, .
Constraint: ${\mathbf{F}}\left(\mathit{j}\right)\ge 0.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{LF}}$.
5:     LDF1 – INTEGERInput
On entry: the length of the array DF1.
Constraint: ${\mathbf{LDF1}}>0$.
6:     DF1(LDF1) – REAL (KIND=nag_wp) arrayInput
On entry: ${u}_{i}$, the degrees of freedom of the numerator variance with ${u}_{i}={\mathbf{DF1}}\left(j\right)$, .
Constraint: ${\mathbf{DF1}}\left(\mathit{j}\right)>0.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{LDF1}}$.
7:     LDF2 – INTEGERInput
On entry: the length of the array DF2.
Constraint: ${\mathbf{LDF2}}>0$.
8:     DF2(LDF2) – REAL (KIND=nag_wp) arrayInput
On entry: ${v}_{i}$, the degrees of freedom of the denominator variance with ${v}_{i}={\mathbf{DF2}}\left(j\right)$, .
Constraint: ${\mathbf{DF2}}\left(\mathit{j}\right)>0.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{LDF2}}$.
9:     P($*$) – REAL (KIND=nag_wp) arrayOutput
Note: the dimension 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($*$) – INTEGER arrayOutput
Note: the dimension 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: ${\mathbf{IVALID}}\left(i\right)$ indicates any errors with the input arguments, with
${\mathbf{IVALID}}\left(i\right)=0$
No error.
${\mathbf{IVALID}}\left(i\right)=1$
 On entry, invalid value supplied in TAIL when calculating ${p}_{i}$.
${\mathbf{IVALID}}\left(i\right)=2$
 On entry, ${f}_{i}<0.0$.
${\mathbf{IVALID}}\left(i\right)=3$
 On entry, ${u}_{i}\le 0.0$, or ${v}_{i}\le 0.0$.
${\mathbf{IVALID}}\left(i\right)=4$
The solution has failed to converge. The result returned should represent an approximation to the solution.
11:   IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, because for this routine the values of the output parameters may be useful even if ${\mathbf{IFAIL}}\ne {\mathbf{0}}$ on exit, the recommended value is $-1$. When the value $-\mathbf{1}\text{​ or ​}1$ is used it is essential to test the value of IFAIL on exit.
On exit: ${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Note: G01SDF may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the routine:
${\mathbf{IFAIL}}=1$
On entry, at least one value of F, DF1, DF2 or TAIL was invalid, or the solution failed to converge.
${\mathbf{IFAIL}}=2$
On entry, $\text{array size}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{LTAIL}}>0$.
${\mathbf{IFAIL}}=3$
On entry, $\text{array size}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{LF}}>0$.
${\mathbf{IFAIL}}=4$
On entry, $\text{array size}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{LDF1}}>0$.
${\mathbf{IFAIL}}=5$
On entry, $\text{array size}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{LDF2}}>0$.
${\mathbf{IFAIL}}=-999$
Dynamic memory allocation failed.

## 7  Accuracy

The result should be accurate to five significant digits.

For higher accuracy G01SEF 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 (g01sdfe.f90)

### 9.2  Program Data

Program Data (g01sdfe.d)

### 9.3  Program Results

Program Results (g01sdfe.r)