Testing mean changes by maximal ratio statistics

Abstract

We propose a new test statistic \(\mathrm {MR}_{\gamma ,n}\) for detecting a changed segment in the mean, at unknown dates, in a regularly varying sample. Our model supports several alternatives of shifts in the mean, including one change point, constant, epidemic and linear form of a change. Our aim is to detect a short length changed segment \(\ell ^{*}\), assuming \(\ell^*/n\) to be small as the sample size n is large. \(\mathrm {MR}_{\gamma ,n}\) is built by taking maximal ratios of weighted moving sums statistics of four sub-samples. An important feature of \(\mathrm {MR}_{\gamma ,n}\) is to be scale free. We obtain the limiting distribution of ratio statistics under the null hypothesis as well as their consistency under the alternative. These results are extended from i.i.d. samples under \(H_0\) to some dependent samples. To supplement theoretical results, empirical illustrations are provided by generating samples from symmetrized Pareto and Log-Gamma distributions.