A Non Parametric Exact Test Based on the Number of Records for an Early Detection of Emerging Events: Illustration in Epidemiology

Facing the first times of occurrence of events of a new type, such as seismic events, arrivals of individuals of a unknown species, occurrences of cases of a new disease, etc., it is crucial to predict if these events are only “sporadic,” or if they announce an emerging phenomenon (earthquake, emergence of a population, epidemic). We propose here an exact non parametric test statistic based on the number of lower records Nn in {ΔTk}1 ⩽ k ⩽ n, ΔTk being the waiting time between two successive events. Under H0 (sporadic events), the {ΔTk} are assumed i.i.d., while under H1 (emergent events), the {ΔTk} are assumed independent with cdf’s (cumulative distributions functions) {Gk} increasing with k. Under H0, the distribution of Nn is independent of Gk, thus allowing a non parametric test of H0. To calculate the power of the test under the alternative hypothesis H1, we assume the particular family of cdf, Gk=1-(1-G)ak$G_k= 1-(1-G)^{a^k}$, k∈N$k\in {\mathbb {N}}$, where G is a continuous cdf on (0, ∞) and a > 1. These distributions represent an exponential occurrence rate of events. We show that the distribution of Nn under H1 depends only on a (and not on G). We estimate a by the maximum likelihood estimator, and give the asymptotic properties of this estimator, as n → ∞. Finally, we illustrate the test on simulations and then on data concerning the atypical bovine spongiform encephalopathy in France.