## The Annals of Statistics

### Equivalence theory for density estimation, Poisson processes and Gaussian white noise with drift

#### Abstract

This paper establishes the global asymptotic equivalence between a Poisson process with variable intensity and white noise with drift under sharp smoothness conditions on the unknown function. This equivalence is also extended to density estimation models by Poissonization. The asymptotic equivalences are established by constructing explicit equivalence mappings. The impact of such asymptotic equivalence results is that an investigation in one of these nonparametric models automatically yields asymptotically analogous results in the other models.

#### Article information

Source
Ann. Statist., Volume 32, Number 5 (2004), 2074-2097.

Dates
First available in Project Euclid: 27 October 2004

https://projecteuclid.org/euclid.aos/1098883782

Digital Object Identifier
doi:10.1214/009053604000000012

Mathematical Reviews number (MathSciNet)
MR2102503

Zentralblatt MATH identifier
1062.62083

Subjects
Primary: 62B15: Theory of statistical experiments
Secondary: 62G07: Density estimation 62G20: Asymptotic properties

#### Citation

Brown, Lawrence D.; Carter, Andrew V.; Low, Mark G.; Zhang, Cun-Hui. Equivalence theory for density estimation, Poisson processes and Gaussian white noise with drift. Ann. Statist. 32 (2004), no. 5, 2074--2097. doi:10.1214/009053604000000012. https://projecteuclid.org/euclid.aos/1098883782

#### References

• Brown, L. D., Cai, T., Low, M. G. and Zhang, C.-H. (2002). Asymptotic equivalence theory for nonparametric regression with random design. Ann. Statist. 30 688--707.
• Brown, L. D. and Low, M. G. (1996). Asymptotic equivalence of nonparametric regression and white noise. Ann. Statist. 24 2384--2398.
• Brown, L. D. and Zhang, C.-H. (1998). Asymptotic nonequivalence of nonparametric experiments when the smoothness index is $1/2$. Ann. Statist. 26 279--287.
• Carter, A. V. (2000). Asymptotic equivalence of nonparametric experiments. Ph.D. dissertation, Yale Univ.
• Carter, A. V. (2002). Deficiency distance between multinomial and multivariate normal experiments. Ann. Statist. 30 708--730.
• Carter, A. V. and Pollard, D. (2004). Tusnády's inequality revisited. Ann. Statist. 32. To appear.
• Efromovich, S. and Samarov, A. (1996). Asymptotic equivalence of nonparametric regression and white noise has its limits. Statist. Probab. Lett. 28 143--145.
• Folland, G. B. (1984). Real Analysis. Wiley, New York.
• Golubev, G. and Nussbaum, M. (1998). Asymptotic equivalence of spectral density and regression estimation. Technical report, Weierstrass Institute, Berlin.
• Gramma, I. and Nussbaum, M. (1998). Asymptotic equivalence for nonparametric generalized linear models. Probab. Theory Related Fields 111 167--214.
• Le Cam, L. (1964). Sufficiency and approximate sufficiency. Ann. Math. Statist. 35 1419--1455.
• Le Cam, L. (1986). Asymptotic Methods in Statistical Decision Theory. Springer, New York.
• Le Cam, L. and Yang, G. (1990). Asymptotics in Statistics. Springer, New York.
• Milstein, G. and Nussbaum, M. (1998). Diffusion approximation for nonparametric autoregression. Probab. Theory Related Fields 112 535--543.
• Nussbaum, M. (1996). Asymptotic equivalence of density estimation and Gaussian white noise. Ann. Statist. 24 2399--2430.