A Multivariate Cumulative Damage Model And Some Applications

A continuous multi-time stochastic model is proposed for a multi-component system, that suffers damage in its components through shocks, occurring at random instants and where the magnitude of the damage produced, by each shock, is random. Central limit theorems are stated for two sequences of martingales, which allow knowing the asymptotic distribution of the cumulative damage of the system at eventually different times depending on its components. Moreover, we introduce large deviation principles for related processes. Thresholds for the components are stated, and the multivariate stopping time limits, where each component's damage attains the corresponding threshold, are studied. The main interest of the paper is focused on introducing some useful tools for statistical inference on the parameters of the system. Cumulative damage is present in many areas of study such as earthquakes, reliability, and finance, among others, and, in this work, some of the asymptotic results obtained are applied to the analysis of infectious diseases. In particular, a hypothesis test for the infection homogeneity of the pandemic COVID 19 in Chile is carried out. This test is applied to real data and some simulations are conducted.