Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 218 / 5 / d / Solution

A weighted nearest-neighbours classifier predicts one whenwhere neighbours are distance ordered. Under the usual smooth-density and smooth-regression assumptions, asymptotically optimal weights downweight distant neighbours, for example normalized positive parts of $1-(i/k{)}^{2/p}$. The optimal weighted-nearest-neighbour theorem gives smaller leading asymptotic regret than equal weights.