NAG Library Function Document

nag_triplets_test (g08ecc)

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

[0, 1]

observations.

m

m

matrix,

C

, of counts is formed as follows. The element

c_{j k l}

C

is the number of triplets (x

(i)

, x

(i + 1)

, x

(i + 2)

), for

i = 1, 4, \dots, n - 2

, such that

\frac{j - 1}{m} \leq X (i) < \frac{j}{m}

\frac{k - 1}{m} \leq X (i + 1) < \frac{k}{m}

\frac{l - 1}{m} \leq X (i + 2) < \frac{l}{m} .

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

{[0, 1]}^{3}

. Thus the expected number of triplets for each class is just the total number of triplets,

\sum_{j, k, l = 1}^{m} c_{j k l}

, divided by the number of classes,

m^{3}

The

χ^{2}

test statistic used to test the hypothesis of randomness is defined as:

X^{2} = \sum_{j, k, l = 1}^{m} \frac{{(c_{j k l} - e)}^{2}}{e}

where

e = \sum_{j, k, l = 1}^{m} c_{j k l} / m^{3} =

expected number of triplets in each class.

The use of the

χ^{2}

distribution as an approximation to the exact distribution of the test statistic,

X^{2}

, improves as the length of the sequence relative to

m

increases, 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 – IntegerInput: On entry: $n$ , the number of observations.
Constraint: $n \geq 3$ .
2: x[n] – const doubleInput: On entry: the sequence of observations.
Constraint: $0.0 \leq x [i - 1] \leq 1.0$ , for $i = 1, 2, \dots, n$ .
3: max_count – IntegerInput: On entry: the size of the count matrix to be formed, $m$ .
Constraint: $max_count \geq 2$ .
4: chi – double *Output: On exit: contains the $χ^{2}$ test statistic, $X^{2}$ , for testing the null hypothesis of randomness.
5: df – double *Output: On exit: contains the degrees of freedom for the $χ^{2}$ statistic.
6: prob – double *Output: On exit: contains the upper tail probability associated with the $χ^{2}$ test statistic, i.e., the significance level.
7: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_G08EC_CELL: 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 $χ^{2}$ distribution may not be a very good approximation to the test statistic.
NE_G08EC_TRIPLETS: No triplets were found because less than 3 observations were provided in total.
NE_INT_ARG_LE: On entry, max_count must not be less than or equal to 1: $max_count = ⟨value⟩$ .
NE_INT_ARG_LT: On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 3$ .
NE_INTERNAL_ERROR: 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.
NE_REAL_ARRAY_CONS: On entry, $x [⟨value⟩] = ⟨value⟩$ .
Constraint: $0 < x [i] < 1.0$ , for $i = 0, 1, \dots, n - 1$ .

7 Accuracy

The computations are believed to be stable. The computations of prob given the values of chi and df will obtain a relative accuracy of five significant figures for most cases.