Asymptotic expansions of the $k$ nearest neighbor risk
k 近邻分类风险的渐近展开
The finite-sample risk of the $k$ nearest neighbor classifier that uses a weighted $L^p$-metric as a measure of class similarity is examined. For a family of classification problems with smooth distributions in $mathbb{R}^n$, an asymptotic expansion for the risk is obtained in decreasing fractional powers of the reference sample size. An analysis of the leading expansion coefficients reveals that the optimal weighted $L^p$-metric, that is, the metric that minimizes the…