Serinaldi, FrancescoFrancescoSerinaldiKilsby, ChrisChrisKilsby2026-03-172026-03-172013-03-01Physica A: Statistical Mechanics and Its Applications 392 (5): 1080-1089 (2013)https://hdl.handle.net/11420/62170The Allan factor (AF) is a statistic widely used to assess if the rate of occurrences of an event tends to cluster and show persistence in a range of space and/or time scales. For a homogeneous Poisson process, the relationship between AF and the space/time is expected to be constant, thus denoting the lack of clustering and persistence in the occurrence process. However, in time series analysis, conclusions about the persistence of the underlying process have been usually drawn by visual inspection of the diagrams of AF estimates versus scale, without applying any formal statistical test. This study investigates the sampling distribution function of the AF estimator when the underlying process is homogeneous Poissonian. Monte Carlo simulations show that the distribution of the AF estimator is described by a gamma distribution whose mean and variance can be deduced by the delta method. Therefore, the derived analytical distribution of the AF estimator can be used to build a formal statistical test to evaluate the significance of the AF fluctuation against the Poissonian hypothesis across a range of space/time scales. As an example, we apply the AF-based test to analyse the time series of the number of rainfall observations exceeding fixed high thresholds in order to study the properties of the rate of occurrence of the extreme values over a wide range of time scales. © 2012 Elsevier B.V. All rights reserved.en0378-43712013510801089ElsevierAllan factorClusteringDispersionFractalsPeak over threshold analysisPoisson processStatistical testNatural Sciences and Mathematics::551: Geology, Hydrology MeteorologyJournal Article10.1016/j.physa.2012.11.015Journal Article