NAG Library Function Document
nag_triplets_test (g08ecc) performs the triplets test on a sequence of observations from the interval .
||nag_triplets_test (Integer n,
const double x,
nag_triplets_test (g08ecc) computes the statistics for performing a triplets test which may be used to investigate deviations from randomness in a sequence of observations.
, of counts is formed as follows. The element
is the number of triplets (x
, such that
Note that all triplets formed are non-overlapping and are thus independent under the assumption of randomness.
Under the assumption that the sequence is random, the expected number of triplets for each class (i.e., each element of the count matrix) is the same, that is the triplets should be uniformly distributed over the unit cube . Thus the expected number of triplets for each class is just the total number of triplets, , divided by the number of classes, .
test statistic used to test the hypothesis of randomness is defined as:
expected number of triplets in each class.
The use of the distribution as an approximation to the exact distribution of the test statistic, , improves as the length of the sequence relative to increases, hence the expected value, , increases.
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
n – IntegerInput
On entry: , the number of observations.
x[n] – const doubleInput
On entry: the sequence of observations.
, for .
max_count – IntegerInput
On entry: the size of the count matrix to be formed, .
chi – double *Output
On exit: contains the test statistic, , for testing the null hypothesis of randomness.
df – double *Output
On exit: contains the degrees of freedom for the statistic.
prob – double *Output
On exit: contains the upper tail probability associated with the test statistic, i.e., the significance level.
fail – NagError *Input/Output
The NAG error argument (see Section 3.6
in the Essential Introduction).
6 Error Indicators and Warnings
Dynamic memory allocation failed.
The expected value for the counts in each element of the count matrix is less than or equal to 5.0. This implies that the distribution may not be a very good approximation to the test statistic.
No triplets were found because less than 3 observations were provided in total.
On entry, max_count
must not be less than or equal to 1:
On entry, .
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
On entry, .
Constraint: , for .
The computations are believed to be stable. The computations of prob
given the values of chi
will obtain a relative accuracy of five significant figures for most cases.
8 Parallelism and Performance
The time taken by nag_triplets_test (g08ecc) increases with the number of observations, .
The following program performs the pairs test on 10000 pseudorandom numbers from a uniform distribution
generated by nag_rand_basic (g05sac)
. nag_triplets_test (g08ecc) is called with max_count
set to 5.
10.1 Program Text
Program Text (g08ecce.c)
10.2 Program Data
10.3 Program Results
Program Results (g08ecce.r)