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

x = \{x_{i} : i = 1, 2, \dots, n\}

, 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}, x_{i + l})

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

, 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,

\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: $n$ – IntegerInput: On entry: $n$ , the number of observations.

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: $m$ , the size of the matrix of counts.

Constraint: $max_count \geq 2$ .
4: $lag$ – IntegerInput: 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 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation 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.

8

Parallelism and Performance

nag_pairs_test (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 nag_rand_basic (g05sac). nag_pairs_test (g08ebc) is called with

lag = 1

and

max_count = 10

10.1

Program Text

Program Text (g08ebce.c)

10.2

Program Data

None.

10.3

Program Results

Program Results (g08ebce.r)

nag_pairs_test (g08ebc) (PDF version)

g08 Chapter Contents

g08 Chapter Introduction

NAG Library Manual

NAG Library Function Document

nag_pairs_test (g08ebc)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification

3

Description