## On the Mutual Nearest Neighbors Estimate in Regression

*Arnaud Guyader, Nick Hengartner*; 14(Aug):2361−2376, 2013.

### Abstract

Motivated by promising experimental results, this paper
investigates the theoretical properties of a recently proposed
nonparametric estimator, called the Mutual Nearest Neighbors
rule, which estimates the regression function
$m(\mathbf{x})=\mathbb E[Y|\mathbf{X}=\mathbf{x}]$ as follows:
first identify the $k$ nearest neighbors of $\mathbf{x}$ in the
sample $\mathcal{D}_n$, then keep only those for which
$\mathbf{x}$ is itself one of the $k$ nearest neighbors, and
finally take the average over the corresponding response
variables. We prove that this estimator is consistent and that
its rate of convergence is optimal. Since the estimate with the
optimal rate of convergence depends on the unknown distribution
of the observations, we also present adaptation results by data-
splitting.

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