Given any countable collection of regression procedures (e.g., kernel, spline, wavelet, local polynomial, neural nets, etc), we show that a single adaptive procedure can be constructed to share the advantages of them to a great extent in terms of global L_2 risk. The combined procedure basically pays a price only of order 1/n for adaptation over the collection. An interesting consequence is that for a countable collection of classes of regression functions (possibly of completely different characteristics), a minimax-rate adaptive estimator can be constructed such that it automatically converges at the right rate for each of the classes being considered.

A demonstration is given for high dimensional regression, for which case, to overcome the well-known curse of dimensionality in accuracy, it is advantageous to seek different ways of characterizing a high-dimensional function (e.g., using neural nets or additive modelings) to reduce the influence of input dimension in the traditional theory of

approximation (e.g., in terms of series expansion). However, in general, it is difficult to assess which characterization works well for the unknown regression function. Thus adaptation over different modelings is desired. For example, we show by combining various regression procedures, a single estimator can be constructed to be minimax-rate adaptive over Besov classes of unknown smoothness and interaction order, to converge at rate o(n^{-1/2}) when the regression function has a neural net representation, and at the same time to be consistent over all bounded regression functions.

AMS 1991 subject classifications. Primary 62G07; secondary 62B10, 62C20, 94A29.

Key words and phrases. Adaptive estimation, combined procedures, minimax rate, nonparametric regression.

Copies of preprints are available from the author upon request. Use the preprint number (top right hand corner of the abstract) and make the request directly to the author, Iowa State University, Department of Statistics, Snedecor Hall, Ames, IA 50011-1210.