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On statistics of Markov step processes: Representation of log-likelihood ratio processes in filtered local models
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  • Published: September 1993

On statistics of Markov step processes: Representation of log-likelihood ratio processes in filtered local models

  • Reinhard Höpfner1 

Probability Theory and Related Fields volume 94, pages 375–398 (1993)Cite this article

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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

$$h^T M_\vartheta ^n - \tfrac{1}{2}h^T \left\langle {M_\vartheta ^n } \right\rangle h + remainder terms,$$

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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Authors and Affiliations

  1. Institut für Mathematische Stochastik, Albert-Ludwigs-Universität Freiburg, Hebelstrasse 27, W-7800, Freiburg im Breisgau, Federal Republic of Germany

    Reinhard Höpfner

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  1. Reinhard Höpfner
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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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  • Received: 10 May 1991

  • Revised: 16 April 1992

  • Issue Date: September 1993

  • DOI: https://doi.org/10.1007/BF01199249

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Mathematics subject classification (1991)

  • 62M05
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