Testing for unspecified periodicities in binary time series

TL;DR

This paper introduces a statistical test to detect unspecified periodicities in binary time series data, using an auxiliary series and Fisher's g test, with proven asymptotic level control and power under certain conditions.

Contribution

It proposes a novel testing procedure for identifying periodicities of unknown length in binary sequences, extending existing methods with theoretical guarantees.

Findings

Test maintains asymptotic level under null hypothesis.

Test has power for most alternative periodicities, especially when period and auxiliary length share divisors.

Applicable to binary data with potential periodic success probabilities.

Abstract

Given independent random variables $Y_1, \ldots, Y_n$ with $Y_i \in \{0,1\}$ we test the hypothesis whether the underlying success probabilities $p_i$ are constant or whether they are periodic with an unspecified period length of $r \ge 2$. The test relies on an auxiliary integer $d$ which can be chosen arbitrarily, using which a new time series of length $d$ is constructed. For this new time series, the test statistic is derived according to the classical $g$ test by Fisher. Under the null hypothesis of a constant success probability it is shown that the test keeps the level asymptotically, while it has power for most alternatives, i.e. typically in the case of $r \ge 3$ and where $r$ and $d$ have common divisors.