Performs a test for randomness.

The type double function is imsls_d_randomness_test.

The probability of a larger chi‑squared statistic for testing the null hypothesis of a uniform distribution.

Function imsls_f_randomness_test performs one of four different tests for randomness. Optional argument IMSLS_RUNS computes statistics for the runs up test. Runs tests are used to test for cyclical trend in sequences of random numbers. If the runs down test is desired, each observation should first be multiplied by -1 to change its sign, and IMSLS_RUNS called with the modified vector of observations.

IMSLS_RUNS first tallies the number of runs up (increasing sequences) of each desired length. For i = 1, ..., r - 1, where r = n_run, runs_count[i] contains the number of runs of length i. runs_count[n_run] contains the number of runs of length n_run or greater. As an example of how runs are counted, the sequence (1, 2, 3, 1) contains 1 run up of length 3, and one run up of length 1.

After tallying the number of runs up of each length, IMSLS_RUNS computes the expected values and the covariances of the counts according to methods given by Knuth (1981, pages 65-67). Let R denote a vector of length n_run containing the number of runs of each length so that the i‑th element of R, ri, contains the count of the runs of length i. Let ΣR denote the covariance matrix of R under the null hypothesis of randomness, and let μR denote the vector of expected values for R under this null hypothesis, then an approximate chi-squared statistic with n_run degrees of freedom is given as

In general, the larger the value of each element of μR, the better the chi-squared approximation.

IMSLS_PAIRS computes the pairs test (or the Good’s serial test) on a hypothesized sequence of uniform (0,1) pseudo-random numbers. The test proceeds as follows. Subsequent pairs (x[i], x[i + pairs_lag]) are tallied into a k × k matrix, where k = n_run. In this tally, element (j, m) of the matrix is incremented, where

where l = pairs_lag, and the notation ⌊ ⌋ represents the greatest integer function, ⌊Y⌋ is the greatest integer less than or equal to Y, where Y is a real number. If l = 1, then i = 1, 3, 5, ..., n ‑ 1. If l > 1, then i = 1, 2, 3, ..., n - l, where n is the total number of pseudo-random numbers input on the current invocation of IMSLS_PAIRS (i.e., n = n_observations).

Given the tally matrix in pairs_count, chi-squared is computed as

where e = Σoij/k2, and oij is the observed count in cell (i, j) (oij = pairs_count[i][j]).

Because pair statistics for the trailing observations are not tallied on any call, the user should call IMSLS_PAIRS with n_observations as large as possible. For pairs_lag < 20 and n_observations = 2000, little power is lost.

IMSLS_DSQUARE computes the d2 test for succeeding quadruples of hypothesized pseudo-random uniform (0, 1) deviates. The d2 test is performed as follows. Let X1, X2, X3, and X4 denote four pseudo-random uniform deviates, and consider

The probability distribution of D2 is given as

when D2 ≤ 1, where π denotes the value of pi. If D2 > 1, this probability is given as

See Gruenberger and Mark (1951) for a derivation of this distribution.

For each succeeding set of 4 pseudo-random uniform numbers input in X, d2 and the cumulative probability of d2 (Pr(D2 ≤ d2)) are computed. The resulting probability is tallied into one of k = n_run equally spaced intervals.

Let n denote the number of sets of four random numbers input (n = the total number of observations/4). Then, under the null hypothesis that the numbers input are random uniform (0, 1) numbers, the expected value for each element in dsquare_count is e = n/k. An approximate chi-squared statistic is computed as

IMSLS_DCUBE computes the triplets test on a sequence of hypothesized pseudo-random uniform(0, 1) deviates. The triplets test is computed as follows:
Each set of three successive deviates, X1, X2, and X3, is tallied into one of m3 equal sized cubes, where m = n_run. Let i = [mX1] + 1, j = [mX2] + 1, and k = [mX3] + 1. For the triplet (X1, X2, X3), dcube_count[i][j][k] is incremented.

Under the null hypothesis of pseudo-random uniform(0, 1) deviates, the m3 cells are equally probable and each has expected value e = n/m3, where n is the number of triplets tallied. An approximate chi-squared statistic is computed as

where oijk = dcube_count[i][j][k].

This example illustrates the use of the runs test on 104 pseudo-random uniform deviates. Since the probability of a larger chi-squared statistic is 0.1872, there is no strong evidence to support rejection of this null hypothesis of randomness.

This example illustrates the calculations of the IMSLS_PAIRS statistics when a random sample of size 104 is used and the pairs_lag is 1. The results are not significant. IMSL function imsls_f_random_uniform (Chapter 12, Random Number Generation) is used in obtaining the pseudo-random deviates.

In this example, 2000 observations generated via IMSL function imsls_f_random_uniform (Chapter 12, Random Number Generation) are input to IMSLS_DSQUARE in one call. In the example, the null hypothesis of a uniform distribution is not rejected.

In this example, 2001 deviates generated by IMSL function imsls_f_random_uniform (Chapter 12, Random Number Generation) are input to IMSLS_DCUBE, and tabulated in 27 equally sized cubes. In the example, the null hypothesis is not rejected.

This example is based on Example 1 to illustrate the use of the IMSLS_IDO optional argument. In this example, imsls_f_randomness_test is called 10 times, with 1000 pseudo-random uniform deviates each time. Since the probability of a larger chi-squared statistic is 0.1872, there is no strong evidence to support rejection of this null hypothesis of randomness.