NAG CL Interface
g08ebc (randtest_​pairs)

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1 Purpose

g08ebc performs a pairs test on a sequence of observations in the interval [0,1].

2 Specification

#include <nag.h>
void  g08ebc (Integer n, const double x[], Integer max_count, Integer lag, double *chi, double *df, double *prob, NagError *fail)
The function may be called by the names: g08ebc, nag_nonpar_randtest_pairs or nag_pairs_test.

3 Description

g08ebc 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×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 [0,1]2. 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.

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 Arguments

1: n Integer Input
On entry: n, the number of observations.
Constraint: n2.
2: x[n] const double Input
On entry: the sequence of observations.
Constraint: 0.0x[i-1]1.0, for i=1,2,,n.
3: max_count Integer Input
On entry: m, the size of the matrix of counts.
Constraint: max_count2.
4: lag Integer Input
On entry: l, the lag to be used in choosing pairs.
If lag=1, then we consider the pairs (x[i-1],x[i]), for i=1,3,,n-1, where n is the number of observations.
If lag>1, then we consider the pairs (x[i-1],x[i+l-1]), for i=1,2,,l,2l+1,2l+2,,3l,4l+1,,n-l, where n is the number of observations.
Constraint: 1lag<n.
5: chi double * Output
On exit: contains the χ2 test statistic, X2, for testing the null hypothesis of randomness.
6: df double * Output
On exit: contains the degrees of freedom for the χ2 statistic.
7: prob double * Output
On exit: contains the upper tail probability associated with the χ2 test statistic, i.e., the significance level.
8: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument value had an illegal value.
max_count 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.
max_count=value, number of pairs =value and expected value =value.
All statistics are returned and may still be of use.
No pairs were found. This will occur if the value of lag is greater than or equal to the total number of observations.
On entry, lag=value and n=value.
Constraint: 1lag<n.
On entry, max_count=value.
Constraint: max_count2.
On entry, n=value.
Constraint: n2
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 7.5 in the Introduction to the NAG Library CL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
On entry, at least one element of x is out of range.
Constraint: 0x[i-1]1, for i=1,2,,n.

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 figures for most cases.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
g08ebc is not threaded in any implementation.

9 Further Comments

The time taken by the function increases with the number of observations n.

10 Example

The following program performs the pairs test on 10000 pseudorandom numbers taken from a uniform distribution U(0,1), generated by g05sac. g08ebc is called with lag=1 and max_count=10..

10.1 Program Text

Program Text (g08ebce.c)

10.2 Program Data


10.3 Program Results

Program Results (g08ebce.r)