Summary
Observe over a long time the trajectory of a Markov step processX=(X t ) t≧0 whose generator depends on an unknown parameter ϑ. Consider the corresponding statistical model locally over small neighbourhoods of some fixed ϑ. We prove a decomposition of log-likelihood ratio processes in the local model, of type
whereh is the local parameter andM n ϑ is the score function martingale, suitably normed and scaled. Weak convergence of filtered local models at ϑ is reduced to a problem of weak convergence of the score function martingale; this can be solved in a variety of cases. As a consequence, we obtain LAQ, LAMN or LAN at ϑ under fixed time sampling as well as under a certain class of “sequential” sampling plans.
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Höpfner, R. On statistics of Markov step processes: Representation of log-likelihood ratio processes in filtered local models. Probab. Th. Rel. Fields 94, 375–398 (1993). https://doi.org/10.1007/BF01199249
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DOI: https://doi.org/10.1007/BF01199249
Mathematics subject classification (1991)
- 62M05