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Quadratic Approximation for Log-Likelihood Ratio Processes

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Modern Problems of Stochastic Analysis and Statistics (MPSAS 2016)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 208))

Abstract

We consider a sequence of general filtered statistical models with a finite-dimensional parameter. It is tacitly assumed that a proper rescaling of the parameter space is already done (so we deal with a local parameter) and also time rescaling is done if necessary. Our first and main purpose is to give sufficient conditions for the existence of certain uniform in time linear–quadratic approximations of log-likelihood ratio processes. Second, we prove general theorems establishing LAN, LAMN and LAQ properties for these models based on these linear–quadratic approximations. Our third purpose is to prove three theorems related to the necessity of the conditions in our main result. These theorems assert that these conditions are necessarily satisfied if (1) an approximation of a much more general form exists and a (necessary) condition of asymptotic negligibility of jumps of likelihood ratio processes holds, or (2) we have LAN property at every moment of time and the limiting models are continuous in time, or (3) we have LAN property, Hellinger processes are asymptotically degenerate at the terminal times, and the condition of asymptotic negligibility of jumps holds.

Esko Valkeila—Deceased

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Notes

  1. 1.

    To be more precise, approximations and limits are assumed to be uniform on all compact parameter subsets. The corresponding properties are often called ULAN (uniform LAN), ULAMN, etc. However, we omit the letter ‘U’ in the notation.

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Acknowledgements

This research was supported by Suomalainen Tiedeakatemia (A.A. Gushchin), by Academy of Finland grants 210465 and 212875 (E. Valkeila), and by the program Tête-à-tête in Russia (Euler International Mathematical Institute; both authors). For the first author, this work has been funded by the Russian Academic Excellence Project ‘5-100’. The first author is deeply grateful to the referees for a number of comments aimed at improving the exposition.

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Authors and Affiliations

  1. Steklov Mathematical Institute, Gubkina 8, 119991, Moscow, Russia

    Alexander Gushchin

  2. Laboratory of Stochastic Analysis and its Applications, National Research University Higher School of Economics, Myasnitskaya 20, 101000, Moscow, Russia

    Alexander Gushchin

  3. Aalto University, P.O. Box 11100, 00076, Aalto, Finland

    Esko Valkeila

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  1. Alexander Gushchin
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  2. Esko Valkeila
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Correspondence to Alexander Gushchin .

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Editors and Affiliations

  1. Department of Statistics and Data Analysis, Higher School of Economics, Moscow, Russia

    Vladimir Panov

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Gushchin, A., Valkeila, E. (2017). Quadratic Approximation for Log-Likelihood Ratio Processes. In: Panov, V. (eds) Modern Problems of Stochastic Analysis and Statistics. MPSAS 2016. Springer Proceedings in Mathematics & Statistics, vol 208. Springer, Cham. https://doi.org/10.1007/978-3-319-65313-6_8

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