NAG Library Function Document

nag_pairs_test (g08ebc)

nag_pairs_test (g08ebc) computes the statistics for performing a pairs test which may be used to investigate deviations from randomness in a sequence of

[0, 1]

observations.

For a given lag,

l \geq 1

, an

m

m

matrix,

C

, of counts is formed as follows. The element

c_{j k}

C

is the number of pairs (x

(i)

(i + 1)

) such that

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

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

where

i = 1, 3, 5, \dots, n - 1

l = 1

, and

i = 1, 2, \dots, l, 2 l + 1, 2 l + 2, \dots, 3 l, 4 l + 1, \dots, n - l

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,

\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 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: the number of observations,

n

Constraint:

n \geq 2

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 matrix of counts,

m

Constraint:

max_count \geq 2

4: lag – IntegerInput

On entry: the lag,

l

, to be used in choosing pairs.

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

lag > 0, ​ 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 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_G08EB_CELL: The expected value for each 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 test statistic.
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, $lag = 〈value〉$ , $n = 〈value〉$ .
Constraint: $1 \leq lag < n$ .
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: $n \geq 2$ .
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.
NE_REAL_ARRAY_CONS: On entry, $x [0] = 〈value〉$ .
Constraint: $0.0 \leq x [i - 1] \leq 1.0$ , for $i = 1, 2, \dots, n - 1$ .

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.