In this work, we develop a unified matrix-level asymptotic calculus for plug-in non-parametric maximum likelihood estimators in finite Markov models. Starting from the asymptotic distribution of the estimated transition matrix, the limiting object is kept in its natural matrix form as a Gaussian random matrix, while the corresponding row-wise vector representation remains immediately available. The main point is that the stochastic constraints of the transition matrix need not be removed by a minimal parametrization: they are carried by the tangent directions and by the covariance structure of the limiting Gaussian matrix, whereas the relevant differentials are computed directly in matrix spaces. A single stochastic calculus theorem gives first-order limit distributions, finite-order developments for sufficiently differentiable functionals, and analytic expansions when the functional is analytic. This provides a common source for asymptotic formulas for matrix powers, stationary characteristics, finite-dimensional curves of Markov characteristics, additive-functional variances, entropy-type quantities and reliability indicators. The resulting covariance operators lead directly to confidence intervals, confidence regions, simultaneous finite-dimensional bands and Wald-type tests. Since the derivations are expressed through matrix products and Kronecker representations rather than coordinate-wise calculations, the method also gives substantial simplifications and, in many cases, computational gains. The second-order terms identify curvature corrections of smooth functionals and provide refined approximations whenever higher-order information is useful.
G. Gavrilopoulos, Samis Trevezas, Irène Votsi· 0 citations
We develop an exact finite-population stochastic framework for SIR epidemics evolving under Markovian switching between intervention regimes. The epidemic state is augmented by a finite phase component, allowing transmission, recovery, and direct immunity-acquisition rates to depend on the active regime. Phase-transition intensities may depend on the current epidemic state, so that policy escalation can react to the number of infectious individuals. Exploiting the monotonicity of the susceptible compartment, we derive level-wise recursions for the joint Laplace--Stieltjes transform and probability generating function of the extinction time and the number of infections generated before extinction. These recursions yield the infection-count distribution, conditional extinction-time transforms, and mixed moments linking epidemic duration and infection burden, while replacing a large global linear system with small phase-level solves. The framework is illustrated using weekly mpox incidence data from Luxembourg. A baseline one-phase SIR model is calibrated by maximum likelihood under a Poisson observation model. The calibrated baseline is then used for conditional comparisons of fixed control regimes, early versus delayed strict intervention, vaccination-supported control, and state-dependent escalation. The results show how switching mechanisms affect both the total number of infected individuals and the extinction time, including their dispersion. Since the switching mechanisms are specified rather than estimated from the intervention history, the results are conditional model-based comparisons rather than estimates of the historical effects of interventions in Luxembourg.
Vasileios E. Papageorgiou, Irène Votsi, Samis Trevezas· 0 citations
