An optimal kk-nearest neighbor for density estimation

A kk-nearest neighbor method, which has been widely applied in machine learning, is a useful tool to obtain statistical inference for an underlying distribution of multi-dimensional data. However, the knowledge on choosing an optimal order for the kk-nearest neighbor is relatively little. This paper proposes an asymptotic distribution for the nearest neighbor statistic. Under some conditions, we find an optimal unbiased density estimate based on a linear combination of nearest neighbors, and it leads to an optimal choice for the order of the kk-nearest neighbor.