Testing for Unspecified Periodicities in Binary Time Series

Given random variables Y1,…,Yn$$ {Y}_1,\dots, {Y}_n $$ with Yi∈{0,1}$$ {Y}_i\in \left\{0,1\right\} $$ we test the hypothesis whether the underlying success probabilities pi$$ {p}_i $$ are constant or whether they are periodic with an unspecified period length of r≥2$$ r\ge 2 $$. The test relies on an auxiliary integer d$$ d $$ which can be chosen arbitrarily, using which a new time series of length d$$ d $$ is constructed. For this new time series, the test statistic is derived according to the classical g$$ g $$ test by Fisher. Under the null hypothesis of a constant success probability and if the sequence (Yi)i≥1$$ {\left({Y}_i\right)}_{i\ge 1} $$ is independent, it is shown that the test keeps the level asymptotically, while it has power for most alternatives, that is, typically in the case of r≥3$$ r\ge 3 $$ and where r$$ r $$ and d$$ d $$ have common divisors. We also discuss extensions to more general distributions than binary ones and prove related limit theorems when the independence assumption is replaced by some weak dependence.