An approximate entropy test for randomness

We develop an entropy-based test for randomness of binary time series of finite length. The test uses the frequencies of contiguous blocks of different lengths. A simple condition on the block lengths and the length of the time series enables one to estimate the entropy rate for the data, and this information is used to develop a statistic to test the hypothesis of randomness. This statistic measures the deviation of the estimated entropy of the observed data from the theoretical maximum under the randomness hypothesis. This test offers a real alternative to the conventional runs test. Critical percentage points, based on simulations, are provided for testing the hypothesis of randomness. Power calculations using dependent data show that the proposed test has higher power against the runs test for short series, and it is similar to the runs test for long series. The test is applied to two published data sets that were investigated by others with respect to their randomness.