NAG Library Routine Document

g08ebf  (randtest_pairs)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{cl}$ – Character(1)Input

On entry: indicates the type of call to g08ebf.

$\mathbf{cl}=\text{'S'}$

This is the one and only call to g08ebf (single call mode). All data are to be input at once. All test statistics are computed after the counting of pairs is complete.

$\mathbf{cl}=\text{'F'}$

This is the first call to the routine. All initializations are carried out and the counting of pairs begins. The final test statistics are not computed since further calls will be made to g08ebf.

$\mathbf{cl}=\text{'I'}$

This is an intermediate call during which the counts of pairs are updated. The final test statistics are not computed since further calls will be made to g08ebf.

$\mathbf{cl}=\text{'L'}$

This is the last call to g08ebf. The test statistics are computed after the final counting of runs is complete.

Constraint: $\mathbf{cl}=\text{'S'}$, $\text{'F'}$, $\text{'I'}$ or $\text{'L'}$.

2:     $\mathbf{n}$ – IntegerInput

On entry: $n$, the number of observations.

Constraints:

• if $\mathbf{cl}=\text{'S'}$, $\mathbf{n}\ge 2$;
• otherwise $\mathbf{n}\ge 1$.

3:     $\mathbf{x}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: the sequence of observations.

Constraint: $0.0\le \mathbf{x}\left(\mathit{i}\right)\le 1.0$, for $\mathit{i}=1,2,\dots ,n$.

4:     $\mathbf{msize}$ – IntegerInput

On entry: $m$, the size of the matrix of counts.

msize must not be changed between calls to g08ebf.

Constraint: $\mathbf{msize}\ge 2$.

5:     $\mathbf{lag}$ – IntegerInput

On entry: $l$, the lag to be used in choosing pairs.

If $\mathbf{lag}=1$, then we consider the pairs $\left(\mathbf{x}\left(\mathit{i}\right),\mathbf{x}\left(\mathit{i}+1\right)\right)$, for $\mathit{i}=1,3,\dots ,n-1$, where $n$ is the number of observations.

If $\mathbf{lag}>1$, then we consider the pairs $\left(\mathbf{x}\left(i\right),\mathbf{x}\left(i+l\right)\right)$, for $i=1,2,\dots ,l,2l+1,2l+2,\dots ,3l,4l+1,\dots ,n-l$, where $n$ is the number of observations.

lag must not be changed between calls to g08ebf.

Constraints:

• $\mathbf{lag}\ge 1$;
• if $\mathbf{cl}=\text{'S'}$, $\mathbf{lag}<\mathbf{n}$.

6:     $\mathbf{ncount}\left(\mathbf{ldc},\mathbf{msize}\right)$ – Integer arrayInput/Output

On entry: if $\mathbf{cl}=\text{'S'}$ or $\text{'F'}$, ncount need not be set.

If $\mathbf{cl}=\text{'I'}$ or $\text{'L'}$, ncount must contain the values returned by the previous call to g08ebf.

On exit: is an msize by msize matrix containing the counts of the number of pairs in each cell, $c_{ij}$, for $\mathit{i}=1,2,\dots ,m$ and $\mathit{j}=1,2,\dots ,m$.

7:     $\mathbf{ldc}$ – IntegerInput

On entry: the first dimension of the array ncount as declared in the (sub)program from which g08ebf is called.

Constraint: $\mathbf{ldc}\ge \mathbf{msize}$.

8:     $\mathbf{ex}$ – Real (Kind=nag_wp)Output

On exit: if $\mathbf{cl}=\text{'S'}$ or $\text{'L'}$ (i.e., if it is a final exit) then ex contains the expected number of counts in each cell, $e$.

Otherwise ex is not set.

9:     $\mathbf{chi}$ – Real (Kind=nag_wp)Output

On exit: if $\mathbf{cl}=\text{'S'}$ or $\text{'L'}$ (i.e., if it is a final exit) then chi contains the ${\chi }^2$ test statistic, $X^2$, for testing the null hypothesis of randomness.

Otherwise chi is not set.

10:   $\mathbf{df}$ – Real (Kind=nag_wp)Output

On exit: if $\mathbf{cl}=\text{'S'}$ or $\text{'L'}$ (i.e., if it is a final exit) then df contains the degrees of freedom for the ${\chi }^2$ statistic.

Otherwise df is not set.

11:   $\mathbf{prob}$ – Real (Kind=nag_wp)Output

On exit: if $\mathbf{cl}=\text{'S'}$ or $\text{'L'}$ (i.e., if it is a final exit) then prob contains the upper tail probability associated with the ${\chi }^2$ test statistic, i.e., the significance level.

Otherwise prob is not set.

12:   $\mathbf{wrk}\left(2\times \mathbf{lag}\right)$ – Real (Kind=nag_wp) arrayCommunication Array

wrk is used to store information between successive calls to g08ebf and therefore must not be changed.

13:   $\mathbf{ifail}$ – IntegerInput/Output

On entry: ifail must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 arguments 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

$\mathbf{ifail}=1$

On entry, $\mathbf{cl}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{cl}=\text{'S'}$, $\text{'F'}$, $\text{'I'}$ or $\text{'L'}$.

$\mathbf{ifail}=2$

On entry, $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: if $\mathbf{cl}=\text{'S'}$, $\mathbf{n}\ge 2$, otherwise $\mathbf{n}\ge 1$.

$\mathbf{ifail}=3$

On entry, $\mathbf{msize}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{msize}\ge 2$ 

$\mathbf{ifail}=4$

On entry, $\mathbf{lag}=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{lag}>0$ and if $\mathbf{cl}=\text{'S'}$, $\mathbf{lag}<\mathbf{n}$.

$\mathbf{ifail}=5$

On entry, $\mathbf{ldc}=〈\mathit{\text{value}}〉$ and $\mathbf{msize}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{ldc}\ge \mathbf{msize}$.

$\mathbf{ifail}=6$

On entry, at least one element of x is out of range.
Constraint: $0\le \mathbf{x}\left(i\right)\le 1$, for $i=1,2,\dots ,\mathbf{n}$.

$\mathbf{ifail}=7$

No pairs were found. This will occur if the value of lag is greater than or equal to the total number of observations.

$\mathbf{ifail}=8$

msize is too large relative to the number of pairs, therefore the expected value for at least one cell is less than or equal to $5.0$.
This implies that the ${\chi }^2$ distribution may not be a very good approximation to the distribution of test statistic.
$\mathbf{msize}=〈\mathit{\text{value}}〉$, number of pairs $=〈\mathit{\text{value}}〉$ and expected value $=〈\mathit{\text{value}}〉$.
All statistics are returned and may still be of use.

$\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-999$

Dynamic memory allocation failed.

See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (g08ebfe.f90)

10.2

Program Data

Program Data (g08ebfe.d)

10.3

Program Results

Program Results (g08ebfe.r)