The Annals of Statistics

Asymptotic equivalence of nonparametric regression and white noise

Lawrence D. Brown and Mark G. Low

Full-text: Open access


The principal result is that, under conditions, to any nonparametric regression problem there corresponds an asymptotically equivalent sequence of white noise with drift problems, and conversely. This asymptotic equivalence is in a global and uniform sense. Any normalized risk function attainable in one problem is asymptotically attainable in the other, with the difference in normalized risks converging to zero uniformly over the entire parameter space. The results are constructive. A recipe is provided for producing these asymptotically equivalent procedures. Some implications and generalizations of the principal result are also discussed.

Article information

Ann. Statist. Volume 24, Number 6 (1996), 2384-2398.

First available in Project Euclid: 16 September 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G07: Density estimation
Secondary: 62G20: Asymptotic properties 62M05: Markov processes: estimation

Risk equivalence local asymptotic minimaxity linear estimators


Brown, Lawrence D.; Low, Mark G. Asymptotic equivalence of nonparametric regression and white noise. Ann. Statist. 24 (1996), no. 6, 2384--2398. doi:10.1214/aos/1032181159.

Export citation