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A continuous Gaussian approximation to a nonparametric regression in two dimensions. (English) Zbl 1098.62042
Summary: Estimating the mean in a nonparametric regression on a two-dimensional regular grid of design points is asymptotically equivalent to estimating the drift of a continuous Gaussian process on the unit square. In particular, we provide a construction of a Brownian sheet process with a drift that is almost the mean function in the nonparametric regression. This can be used to apply estimation or testing procedures from the continuous process to the regression experiment as in L. Le Cam’s theory of equivalent experiments [see “Asymptotic methods in statistical decision theory.” (1986; Zbl 0605.62002)]. Our result is motivated by first looking at the amount of information lost in binning the data in a density estimation problem.

MSC:
62G08 Nonparametric regression and quantile regression
62M09 Non-Markovian processes: estimation
62G20 Asymptotic properties of nonparametric inference
Software:
AS 176
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References:
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