NAG Library Routine Document

g08ebf (randtest_pairs)


    1  Purpose
    7  Accuracy


g08ebf performs a pairs test on a sequence of observations in the interval 0,1.


Fortran Interface
Subroutine g08ebf ( cl, n, x, msize, lag, ncount, ldc, ex, chi, df, prob, wrk, ifail)
Integer, Intent (In):: n, msize, lag, ldc
Integer, Intent (Inout):: ncount(ldc,msize), ifail
Real (Kind=nag_wp), Intent (In):: x(n)
Real (Kind=nag_wp), Intent (Inout):: wrk(2*lag)
Real (Kind=nag_wp), Intent (Out):: ex, chi, df, prob
Character (1), Intent (In):: cl
C Header Interface
#include nagmk26.h
void  g08ebf_ (const char *cl, const Integer *n, const double x[], const Integer *msize, const Integer *lag, Integer ncount[], const Integer *ldc, double *ex, double *chi, double *df, double *prob, double wrk[], Integer *ifail, const Charlen length_cl)


g08ebf computes the statistics for performing a pairs test which may be used to investigate deviations from randomness in a sequence, x=xi:i=1,2,,n, of 0,1 observations.
For a given lag, l1, an m by m matrix, C, of counts is formed as follows. The element cjk of C is the number of pairs xi,xi+l such that
k- 1mxi+l<km  
where i=1,3,5,,n-1 if l=1, and i=1,2,,l,2l+1,2l+2,3l,4l+1,,n-l, if l>1.
Note that all pairs formed are non-overlapping pairs and are thus independent under the assumption of randomness.
Under the assumption that the sequence is random, the expected number of pairs for each class (i.e., each element of the matrix of counts) is the same; that is, the pairs should be uniformly distributed over the unit square 0,12. Thus the expected number of pairs for each class is just the total number of pairs, j,k=1mcjk, divided by the number of classes, m2.
The χ2 test statistic used to test the hypothesis of randomness is defined as
X2=j,k=1m cjk-e 2e,  
where e=j,k=1mcjk/m2= expected number of pairs in each class.
The use of the χ2-distribution as an approximation to the exact distribution of the test statistic, X2, improves as the length of the sequence relative to m increases and hence the expected value, e, increases.
g08ebf may be used in two different modes:
(i) a single call to g08ebf which computes all test statistics after counting the pairs;
(ii) multiple calls to g08ebf with the final test statistics only being computed in the last call.
The second mode is necessary if all the data do not fit into the memory. See argument cl in Section 5 for details on how to invoke each mode.


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


1:     cl – Character(1)Input
On entry: indicates the type of call to g08ebf.
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.
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.
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.
This is the last call to g08ebf. The test statistics are computed after the final counting of runs is complete.
Constraint: cl='S', 'F', 'I' or 'L'.
2:     n – IntegerInput
On entry: n, the number of observations.
  • if cl='S', n2;
  • otherwise n1.
3:     xn – Real (Kind=nag_wp) arrayInput
On entry: the sequence of observations.
Constraint: 0.0xi1.0, for i=1,2,,n.
4:     msize – IntegerInput
On entry: m, the size of the matrix of counts.
msize must not be changed between calls to g08ebf.
Constraint: msize2.
5:     lag – IntegerInput
On entry: l, the lag to be used in choosing pairs.
If lag=1, then we consider the pairs xi,xi+1, for i=1,3,,n-1, where n is the number of observations.
If lag>1, then we consider the pairs xi,xi+l, for i=1,2,,l,2l+1,2l+2,,3l,4l+1,,n-l, where n is the number of observations.
lag must not be changed between calls to g08ebf.
  • lag1;
  • if cl='S', lag<n.
6:     ncountldcmsize – Integer arrayInput/Output
On entry: if cl='S' or 'F', ncount need not be set.
If cl='I' or '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, cij, for i=1,2,,m and j=1,2,,m.
7:     ldc – IntegerInput
On entry: the first dimension of the array ncount as declared in the (sub)program from which g08ebf is called.
Constraint: ldcmsize.
8:     ex – Real (Kind=nag_wp)Output
On exit: if cl='S' or '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:     chi – Real (Kind=nag_wp)Output
On exit: if cl='S' or 'L' (i.e., if it is a final exit) then chi contains the χ2 test statistic, X2, for testing the null hypothesis of randomness.
Otherwise chi is not set.
10:   df – Real (Kind=nag_wp)Output
On exit: if cl='S' or 'L' (i.e., if it is a final exit) then df contains the degrees of freedom for the χ2 statistic.
Otherwise df is not set.
11:   prob – Real (Kind=nag_wp)Output
On exit: if cl='S' or 'L' (i.e., if it is a final exit) then prob contains the upper tail probability associated with the χ2 test statistic, i.e., the significance level.
Otherwise prob is not set.
12:   wrk2×lag – Real (Kind=nag_wp) arrayCommunication Array
wrk is used to store information between successive calls to g08ebf and therefore must not be changed.
13:   ifail – IntegerInput/Output
On entry: ifail must be set to 0, -1​ 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​ 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 ifail0 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).

Error Indicators and Warnings

If on entry ifail=0 or -1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Note: g08ebf may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the routine:
On entry, cl=value.
Constraint: cl='S', 'F', 'I' or 'L'.
On entry, n=value.
Constraint: if cl='S', n2, otherwise n1.
On entry, msize=value.
Constraint: msize2 
On entry, lag=value and n=value.
Constraint: lag>0 and if cl='S', lag<n.
On entry, ldc=value and msize=value.
Constraint: ldcmsize.
On entry, at least one element of x is out of range.
Constraint: 0xi1, for i=1,2,,n.
No pairs were found. This will occur if the value of lag is greater than or equal to the total number of observations.
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 χ2 distribution may not be a very good approximation to the distribution of test statistic.
msize=value, number of pairs =value and expected value =value.
All statistics are returned and may still be of use.
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.
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.
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.


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 figures for most cases.

Parallelism and Performance

g08ebf is not thread safe and should not be called from a multithreaded user program. Please see Section 3.12.1 in How to Use the NAG Library and its Documentation for more information on thread safety.
g08ebf is not threaded in any implementation.

Further Comments

If after forming the pairs in an initial or intermediate call to g08ebf there is an observation left over at the end of the sequence, this observation is used at the beginning of the new sequence provided by the following call to g08ebf. Clearly an observation left over from an only or final call to g08ebf is ignored.
The time taken by the routine increases with the number of observations n, and also depends to some extent on whether the call to g08ebf is an only, first, intermediate or last call.


The following program performs the pairs test on 500 pseudorandom numbers. g08ebf is called 5 times with 100 observations on each call. lag=1 is used and the pairs are tallied into a 5 by 5 matrix.

Program Text

Program Text (g08ebfe.f90)

Program Data

Program Data (g08ebfe.d)

Program Results

Program Results (g08ebfe.r)

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