Relative stability of global errors of nonparametric function estimators

This paper presents relative stability properties of various nonparametric density estimators (histogram, kernel estimates) and of regression estimators (partitioning, kernel, and nearest neighbor estimates). In density estimation, let En denote the L₁ error of an estimate calculated from n data, whereas in regression estimation, the L₂ error of the estimate is used. Sufficient conditions for E_n/E{E_n}→1 in probability are provided. If this limit holds, the asymptotic behavior of the random error E_n can be characterized by its expectation E{E_n},, and one may apply, for example, the established rate-of-convergence results for E{En}.