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. (English) Zbl 1062.62083 Ann. Stat. 32, No. 5, 2074-2097 (2004). Summary: 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. Cited in 35 Documents MSC: 62G20 Asymptotic properties of nonparametric inference 62B15 Theory of statistical experiments 62G07 Density estimation 62M05 Markov processes: estimation; hidden Markov models Keywords:decision theory; local limit theorem; quantile transform; white noise model × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] 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. · Zbl 1029.62044 · doi:10.1214/aos/1028674838 [2] Brown, L. D. and Low, M. G. (1996). Asymptotic equivalence of nonparametric regression and white noise. Ann. Statist. 24 2384–2398. · Zbl 0867.62022 · doi:10.1214/aos/1032181159 [3] 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. · Zbl 0932.62061 · doi:10.1214/aos/1030563986 [4] Carter, A. V. (2000). Asymptotic equivalence of nonparametric experiments. Ph.D. dissertation, Yale Univ. [5] Carter, A. V. (2002). Deficiency distance between multinomial and multivariate normal experiments. Ann. Statist. 30 708–730. · Zbl 1029.62005 · doi:10.1214/aos/1028674839 [6] Carter, A. V. and Pollard, D. (2004). Tusnády’s inequality revisited. Ann. Statist. 32 . · Zbl 1076.62012 · doi:10.1214/009053604000000733 [7] Efromovich, S. and Samarov, A. (1996). Asymptotic equivalence of nonparametric regression and white noise has its limits. Statist. Probab. Lett. 28 143–145. · Zbl 0849.62023 · doi:10.1016/0167-7152(95)00109-3 [8] Folland, G. B. (1984). Real Analysis . Wiley, New York. · Zbl 0549.28001 [9] Golubev, G. and Nussbaum, M. (1998). Asymptotic equivalence of spectral density and regression estimation. Technical report, Weierstrass Institute, Berlin. [10] Gramma, I. and Nussbaum, M. (1998). Asymptotic equivalence for nonparametric generalized linear models. Probab. Theory Related Fields 111 167–214. · Zbl 0953.62039 · doi:10.1007/s004400050166 [11] Le Cam, L. (1964). Sufficiency and approximate sufficiency. Ann. Math. Statist. 35 1419–1455. · Zbl 0129.11202 · doi:10.1214/aoms/1177700372 [12] Le Cam, L. (1986). Asymptotic Methods in Statistical Decision Theory . Springer, New York. · Zbl 0605.62002 [13] Le Cam, L. and Yang, G. (1990). Asymptotics in Statistics . Springer, New York. · Zbl 0719.62003 [14] Milstein, G. and Nussbaum, M. (1998). Diffusion approximation for nonparametric autoregression. Probab. Theory Related Fields 112 535–543. · Zbl 1053.62556 · doi:10.1007/s004400050199 [15] Nussbaum, M. (1996). Asymptotic equivalence of density estimation and Gaussian white noise. Ann. Statist. 24 2399–2430. · Zbl 0867.62035 · doi:10.1214/aos/1032181160 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.