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,

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

, divided by the number of classes,

m^{2}

The

χ^{2}

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

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

where

e = \sum_{j, k = 1}^{m} c_{j k} / m^{2} =

expected number of pairs 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 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: $n \geq 2$ .
2: $x [n]$ – const double Input: On entry: the sequence of observations.

Constraint: $0.0 \leq x [i - 1] \leq 1.0$ , for $i = 1, 2, \dots, n$ .
3: $max_count$ – Integer Input: On entry: $m$ , the size of the matrix of counts.

Constraint: $max_count \geq 2$ .
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 $i = 1, 3, \dots, n - 1$ , where $n$ is the number of observations.

If $lag > 1$ , then we consider the pairs $(x [i - 1], x [i + l - 1])$ , for $i = 1, 2, \dots, l, 2 l + 1, 2 l + 2, \dots, 3 l, 4 l + 1, \dots, n - l$ , where $n$ is the number of observations.

Constraint: $1 \leq lag < n$ .
5: $chi$ – double * Output: On exit: contains the $χ^{2}$ test statistic, $X^{2}$ , for testing the null hypothesis of randomness.
6: $df$ – double * Output: On exit: contains the degrees of freedom for the $χ^{2}$ statistic.
7: $prob$ – double * Output: On exit: contains the upper tail probability associated with the $χ^{2}$ 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

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_G08EB_CELL: 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 $χ^{2}$ 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.
NE_G08EB_PAIRS: No pairs were found. This will occur if the value of lag is greater than or equal to the total number of observations.
NE_INT_2: On entry, $lag = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $1 \leq lag < n$ .
NE_INT_ARG_LE: On entry, $max_count = ⟨ value ⟩$ .
Constraint: $max_count \geq 2$ .
NE_INT_ARG_LT: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 2$
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: 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.
NE_REAL_ARRAY_CONS: On entry, at least one element of x is out of range.
Constraint: $0 \leq x [i - 1] \leq 1$ , for $i = 1, 2, \dots, n$ .

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.