This paper addresses the problem of testing the hypothesis that an observed series is difference stationary. The alternative hypothesis is that the series is another nonstationary process; in particular, an autoregressive model with a random parameter is used. A locally best invariant test is developed assuming Gaussianity, and a representation of its asymptotic distribution as a mixture of Brownian motions is found. The performance of the test in finite samples is investigated by simulation. An example is given where the difference stationary assumption for a well-known data series is rejected.
"Testing a Time Series for Difference Stationarity." Ann. Statist. 23 (3) 1015 - 1028, June, 1995. https://doi.org/10.1214/aos/1176324634