## The Annals of Statistics

### Asymptotic equivalence of estimating a Poisson intensity and a positive diffusion drift

#### Abstract

We consider a diffusion model of small variance type with positive drift density varying in a nonparametric set. We investigate Gaussian and Poisson approximations to this model in the sense of asymptotic equivalence of experiments. It is shown that observation of the diffusion process until its first hitting time of level one is a natural model for the purpose of inference on the drift density. The diffusion model can be discretized by the collection of level crossing times for a uniform grid of levels. The random time increments are asymptotically sufficient and obey a nonparametric regression model with independent data. This decoupling is then used to establish asymptotic equivalence to Gaussian signal-in-white-noise and Poisson intensity models on the unit interval, and also to an i.i.d. model when the diffusion drift function $f$ is a probability density. As an application, we find the exact asymptotic minimax constant for estimating the diffusion drift density with sup-norm loss.

#### Article information

Source
Ann. Statist., Volume 30, Number 3 (2002), 731-753.

Dates
First available in Project Euclid: 6 August 2002

https://projecteuclid.org/euclid.aos/1028674840

Digital Object Identifier
doi:10.1214/aos/1028674840

Mathematical Reviews number (MathSciNet)
MR1922540

Zentralblatt MATH identifier
1029.62071

Subjects
Primary: 62B15: Theory of statistical experiments
Secondary: 62M05: Markov processes: estimation 62G07: Density estimation

#### Citation

Genon-Catalot, Valentine; Laredo, Catherine; Nussbaum, Michael. Asymptotic equivalence of estimating a Poisson intensity and a positive diffusion drift. Ann. Statist. 30 (2002), no. 3, 731--753. doi:10.1214/aos/1028674840. https://projecteuclid.org/euclid.aos/1028674840

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