NAG Library Routine Document

G08EDF

(i)	a single call to G08EDF which computes all test statistics after counting the gaps;
(ii)	multiple calls to G08EDF with the final test statistics only being computed in the last call.

The second mode is necessary if all the data does not fit into the memory. See parameter CL in Section 5 for details on how to invoke each mode.

The term gap is used to describe the distance between two numbers in the sequence that lie in the interval

(r_{l}, r_{u})

. That is, a gap ends at

x_{j}

r_{l} \leq x_{j} \leq r_{u}

. The next gap then begins at

x_{j + 1}

. The interval

(r_{l}, r_{u})

should lie within the region of all possible numbers. For example if the test is carried out on a sequence of

(0, 1)

random numbers then the interval

(r_{l}, r_{u})

must be contained in the whole interval

(0, 1)

. Let

t_{len}

be the length of the interval which specifies all possible numbers.

G08EDF counts the number of gaps of different lengths. Let

c_{i}

denote the number of gaps of length

i

, for

i = 1, 2, \dots, k - 1

. The number of gaps of length

k

or greater is then denoted by

c_{k}

. An unfinished gap at the end of a sequence is not counted unless the sequence is part of an initial or intermediate call to G08EDF (i.e., unless there is another call to G08EDF to follow) in which case the unfinished gap is used. The following is a trivial example.

Suppose we called G08EDF twice (i.e., with

CL ='F'

and then with

CL ='L'

) with the following two sequences and with

RLO = 0.30

and

RUP = 0.60

( $0.20$ $0.40$ $0.45$ $0.40$ $0.15$ $0.75$ $0.95$ $0.23$ ) and
( $0.27$ $0.40$ $0.25$ $0.10$ $0.34$ $0.39$ $0.61$ $0.12$ ).

Then after the second call G08EDF would have counted the gaps of the following lengths:

2, $1$ , $1$ , $6$ , $3$ and $1$ .

When the counting of gaps is complete G08EDF computes the expected values of the counts. An approximate

χ^{2}

statistic with

k

degrees of freedom is computed where

X^{2} = \frac{\sum_{i = 1}^{k} {(c_{i} - e_{i})}^{2}}{e_{i}},

where

$e_{i} = ngaps \times p \times {(1 - p)}^{i - 1}$ , if $i < k$ ;
$e_{i} = ngaps \times {(1 - p)}^{i - 1}$ , if $i = k$ ;
$ngaps =$ the number of gaps found and
$p = (r_{u} - r_{l}) / t_{len}$ .

The use of the

χ^{2}

-distribution as an approximation to the exact distribution of the test statistic improves as the expected values increase.

You may specify the total number of gaps to be found. If the specified number of gaps is found before the end of a sequence G08EDF will exit before counting any further gaps.

4 References

Dagpunar J (1988) Principles of Random Variate Generation Oxford University Press

Knuth D E (1981) The Art of Computer Programming (Volume 2) (2nd Edition) Addison–Wesley

Morgan B J T (1984) Elements of Simulation Chapman and Hall

Ripley B D (1987) Stochastic Simulation Wiley

5 Parameters

1: CL – CHARACTER(1)Input

On entry: indicates the type of call to G08EDF.

$CL ='S'$: This is the one and only call to G08EDF (single call mode). All data are to be input at once. All test statistics are computed after the counting of gaps is complete.
$CL ='F'$: This is the first call to the routine. All initializations are carried out before the counting of gaps begins. The final test statistics are not computed since further calls will be made to G08EDF.
$CL ='I'$: This is an intermediate call during which the counts of gaps are updated. The final test statistics are not computed since further calls will be made to G08EDF.
$CL ='L'$: This is the last call to G08EDF. The test statistics are computed after the final counting of gaps is complete.

Constraint:

CL ='S'

'F'

'I'

'L'

2: N – INTEGERInput

On entry:

n

, the length of the current sequence of observations.

Constraint:

N \geq 1

3: X(N) – REAL (KIND=nag_wp) arrayInput

On entry: the sequence of observations.

4: M – INTEGERInput

On entry: the maximum number of gaps to be sought. If

M \leq 0

then there is no limit placed on the number of gaps that are found.

M should not be changed between calls to G08EDF.

Constraint: if

CL ='S'

M \leq N

5: MAXG – INTEGERInput

On entry:

k

, the length of the longest gap for which tabulation is desired.

MAXG must not be changed between calls to G08EDF.

Constraints:

$MAXG > 1$ ;
if $CL ='S'$ , $MAXG \leq N$ .

6: RLO – REAL (KIND=nag_wp)Input

On entry: the lower limit of the interval to be used to define the gaps,

r_{l}

RLO must not be changed between calls to G08EDF.

7: RUP – REAL (KIND=nag_wp)Input

On entry: the upper limit of the interval to be used to define the gaps,

r_{u}

RUP must not be changed between calls to G08EDF.

Constraint:

RUP > RLO

8: TOTLEN – REAL (KIND=nag_wp)Input

On entry: the total length of the interval which contains all possible numbers that may arise in the sequence.

Constraint:

TOTLEN > 0.0

and

RUP - RLO < TOTLEN

9: NGAPS – INTEGERInput/Output

On entry: if

CL ='S'

'F'

, NGAPS need not be set.

CL ='I'

'L'

, NGAPS must contain the value returned by the previous call to G08EDF.

On exit: the number of gaps actually found,

ngaps

10: NCOUNT(MAXG) – INTEGER arrayInput/Output

On entry: if

CL ='S'

'F'

, NCOUNT need not be set.

CL ='I'

'L'

, NCOUNT must contain the values returned by the previous call to G08EDF.

On exit: the counts of gaps of the different lengths,

c_{i}

, for

i = 1, 2, \dots, k

11: EX(MAXG) – REAL (KIND=nag_wp) arrayOutput

On exit: if

CL ='S'

'L'

(i.e., if it is a final exit) then EX contains the expected values of the counts.

Otherwise the elements of EX are not set.

12: CHI – REAL (KIND=nag_wp)Output

On exit: if

CL ='S'

'L'

(i.e., if it is a final exit) then CHI contains the

χ^{2}

test statistic,

X^{2}

, for testing the null hypothesis of randomness.

Otherwise CHI is not set.

13: DF – REAL (KIND=nag_wp)Output

On exit: if

CL ='S'

'L'

(i.e., if it is a final exit) then DF contains the degrees of freedom for the

χ^{2}

statistic.

Otherwise DF is not set.

14: PROB – REAL (KIND=nag_wp)Output

On exit: if

CL ='S'

'L'

(i.e., if it is a final exit) then PROB contains the upper tail probability associated with the

χ^{2}

test statistic, i.e., the significance level.

Otherwise PROB is not set.

15: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ 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 ​ 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

IFAIL \neq 0

on exit, the recommended value is

- 1

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Note: G08EDF may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the routine:

$IFAIL = 1$

On entry,

CL \neq'S'

'F'

'I'

'L'

$IFAIL = 2$

On entry,

N < 1

$IFAIL = 3$

On entry,

with

CL ='S'

M > N

$IFAIL = 4$

On entry,	$MAXG \leq 1$ ,
or	with $CL ='S'$ , $MAXG > N$ .

$IFAIL = 5$

On entry,	$RLO \geq RUP$ ,
or	$TOTLEN \leq 0.0$ ,
or	$RUP - RLO \geq TOTLEN$ .

$IFAIL = 6$: No gaps were found. You may need to use a longer sequence or increase the size of the interval $(r_{l}, r_{u})$ .

$IFAIL = 7$: The expected frequency of a certain class is zero, that is $e_{i} = 0$ , for some $i = 1, 2, \dots, k$ .

$IFAIL = 8$: The number of gaps requested were not found.

$IFAIL = 9$: Some classes have expected frequencies less than $1.0$ . This implies that the $χ^{2}$ -distribution may not be a very good approximation to the distribution of the test statistic.

7 Accuracy

The computations are believed to be stable. The computation of PROB given the values of CHI and DF will obtain a relative accuracy of five significant places for most cases.

8 Further Comments

The time taken by G08EDF increases with the number of observations

n

, and depends to some extent whether the call is an only, first, intermediate or last call.

9 Example

The following program performs the gaps test on

500

pseudorandom numbers. G08EDF is called

5

times with

100

observations on each call. All gaps of length

10

or more are counted together.

NAG Library Routine DocumentG08EDF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

G08EDF