Laboratoire de Mathématiques Raphaël Salem

UMR 6085 CNRS - Université de Rouen Normandie

We consider truncated detection problems for statistical models with dependent observations given by Bayesian Markov chains for a uniform prior distribution when the number of observations is limited by some known value.
To do this, using optimal stopping methods, a new optimal sequential detection procedure is constructed that minimizes the average delay time in the class of sequential procedures with false positive probabilities not exceeding some fixed value. The main difference between the proposed detection procedure and the usual ones is that it is based not on the posterior probabilities, but on Shiryaev - Roberts statistics. This makes it possible to provide optimal detection in a non-asymptotic sense over a bounded time interval without using additional unknown parameters, in contrast to the well-known Bayesian procedures based on a priori geometric distribution containing an unknown parameter. Then we apply the constructed procedures to the problem of early detection of the beginning of the spread of the epidemic. To this end we use two epidemic models: the binomial models proposed by Baron, Choudhary, Yu (2013) and  the models based on the Gaussian approximation introduced by Pergamenchtchikov, Tartakovsky, Spivak (2022). The obtained theoretical results are confirmed by numerical simulations through the Monte Carlo method.