# SFB 303 Discussion Paper No. A - 144

**Author**: Carroll, R. J., and W. Härdle

**Title**: Symmetrized Nearest Neighbor Regression Estimates

**Abstract**: We consider univariate nonparametric regression. Two standard nonparametric regression function
estimates are kernel estimates and nearest neighbor estimates. Mack (1981) noted that both methods can be
defined with respect to a kernel or weighting function, and that for a given kernel and a suitable choice of
bandwidth, the optimal mean squared error is the same asymptotically for kernel and nearest neighbor estimates.
Yang (1981) defined a new type of nearest neighbor regression estimate using the empirical distribution function
of the predictors to define the window over which to average. This has the effect of forcing the number of
neighbors to be the same both above and below the value of the predictor of interest; we call these symmetrized
nearest neighbor estimates. This estimate is a kernel regression estimate with "predictors" given by the empirical
distribution function of the true predictors. We show that for estimating the regression function at a point, the
optimum mean squared error of this estimate differs from that of the optimum mean squared error for kernel and
ordinary nearest neighbor estimates. No estimate dominates the others. They are asymptotically equivalent with
respect to mean squared error if one is estimating the regression function at a mode of the predictor.

**Keywords**: nonparametric regression, kernel regression, nearest neighbor regression, bias, mean squared error

**JEL-Classification-Number**:

**Creation-Date**: November 1987

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