Minification Integer-Valued Split-BREAK Process with Power Series Innovations and Application in Fire Safety Dynamics

This manuscript introduces a new class of count time series models, referred to as the minification integer-valued Split-BREAK (MIN–SB) process. The proposed framework extends the Split-BREAK modeling philosophy to the integer-valued setting and provides a flexible mechanism for capturing rare events, zero inflation, and structural regime changes frequently observed in safety-related data. The main stochastic properties of the MIN–SB process are derived, including stationarity conditions, explicit moment structure, and correlation dynamics. A key theoretical result reveals an implicit hidden Markov structure underlying the observable process, providing a structural explanation for zero clustering observed in rare-event count processes. Parameter estimation is developed using a simulated method of moments (SMM) approach based on zero-related statistics, and the asymptotic properties of the resulting estimators are established. A Monte Carlo simulation study demonstrates favorable finite-sample performance of the proposed estimation procedure. The practical usefulness of the model is illustrated through an empirical application to time series of injuries and fatalities caused by fire accidents in Serbia. The results show that the MIN–SB specification provides a flexible and accurate framework for modeling zero-inflated count processes arising in fire safety dynamics.