Asymptotic equivalence theory for nonparametric regression with random design

This paper establishes the global asymptotic equivalence between the nonparametric regression with random design and the white noise under sharp smoothness conditions on an unknown regression or drift function. The asymptotic equivalence is established by constructing explicit equivalence mappings between the nonparametric regression and the whitenoise experiments, which provide synthetic observations and synthetic asymptotic solutions from any one of the two experiments with asymptotic properties identical to the true observations and given asymptotic solutions from the other. The impact of such asymptotic equivalence results is that an investigation in one nonparametric problem automatically yields asymptotically analogous results in all other asymptotically equivalent nonparametric problems. 1. Introduction. The purpose of this paper is to establish the global asymptotic equivalence between the nonparametric regression with random design and the white noise under sharp smoothness conditions on an unknown regression or drift function. We establish this asymptotic equivalence by constructing explicit equivalence mappings between the nonparametric regression and the white-noise problems, as in Brown and Low (1996) for their asymptotic equivalence results. The equivalence mapping from the nonparametric regression to the white noise provides synthetic observations of the white noise from the nonparametric regression such that the distributions of the synthetic observations are asymptotically equivalent to those of the true observations of the white noise. For any asymptotic solution to a white-noise problem, the application of the solution to the synthetic observations provides an asymptotic solution to the corresponding nonparametric regression problem with identical asymptotic properties. Likewise, the equivalence mapping from the white noise produces synthetic observations of the nonparametric regression problem and synthetic asymptotic solutions to white-noise problems based on those of the corresponding nonparametric regression problems. The impact of such asymptotic equivalence results is that an investigation in one nonparametric problem automatically yields asymptotically analogous results in