Estimating discontinuous periodic signals in a time inhomogeneous diffusion

By R. Höpfner, Yury Kutoyants. 2009

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We consider a diffusion $${(\xi_t)_{t\geq 0}}$$ with some T-periodic time dependent input term contained in the drift: under an unknown parameter $${\vartheta\in\varTheta}$$ , some discontinuity—an additional periodic signal—occurs at times $${kT\,{+}\,\vartheta}$$ , $${k \in I\!\!N}$$ . Assuming positive Harris recurrence of $${(\xi_{kT})_{k \in I\!\!N _0}}$$ and exploiting the periodicity structure, we prove limit theorems for certain martingales and functionals of the process $${(\xi_t)_{t\ge 0}}$$ . They allow to consider the statistical model parametrized by $${\vartheta\in\varTheta}$$ locally in small neighbourhoods of some fixed $${\vartheta}$$, with radius $${\frac{1}{n}}$$ as n → ∞. We prove convergence of local models to a limit experiment studied by Ibragimov and Khasminskii (Statistical estimation, 1981) and discuss the behaviour of estimators under contiguous alternatives.

Subfields: Computer Networks and Communications, Molecular Biology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/6d744922a986fdabab59b9ff831cbe0a0437940e