WIAS Preprint No. 413, (1998)

A diffusion representation of the nonparametric iid experiment on a interval


  • Genon-Catalot, Valentine
  • Larédo, Catherine
  • Nussbaum, Michael

2010 Mathematics Subject Classification

  • 62B 62M05 62G07


  • Nonparametric experiments, asymptotic equivalence, diffusion process, discretization, level crossing inverse Gaussian regression, signal in white noise, iid model, density estimation




We consider a diffusion model of small variance type with positive drift function varying 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 the collection of level crossing times for a uniform grid of levels is asymptotically equivalent to the continuous model in the sense of Le Cam's deficiency distance, when the discretization step decreases with the noise intensity ε. It follows that in the continuous diffusion model, the statistic of level crossing times is asymptotically sufficient. Since the level crossing times obey a nonparametric regression model with independent data, a further asymptotic equivalence can be established, leading to a simple Gaussian signal-in-white noise problem. When the drift density ƒ is also a probability density, this in turn is asymptotically equivalent to i.i.d. data with density ƒ on the unit interval.

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