WIAS Preprint No. 78, (1993)

Autoregression approximation of a nonparametric diffusion model


  • Milstein, Grigori N.
  • Nussbaum, Michael

2010 Mathematics Subject Classification

  • 62M10 62G07 62B15 60J60 60H10


  • Nonparametric experiments, deficiency distance, likelihood ratio process, stochastic differential equation, autoregression, diffusion sampling, asymptotic sufficiency




We consider a model of small diffusion type where the function which governs the drift term varies in a nonparametric set. We investigate discrete versions of this continuous model with respect to statistical equivalence, in the sense of the asymptotic theory of experiments. It is shown that an Euler difference scheme as a discrete version of the stochastic differential equation is asymptotically equivalent in the sense of Le Cam's deficiency distance, when the discretization step decreases with the noise intensity ∈. We thus obtain a nonparametric version of diffusion limit results for autoregression. It follows that in the continuous diffusion model, discrete sampling on a uniform grid is asymptotically sufficient. The key technical step utilizes the notion of Hellinger process from semimartingale theory.

Appeared in

  • Prob. Th. rel. Fields vol. 112 (1998) no. 4 pp. 535-543 under new title: Diffusion approximation for nonparametric autoregression.

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