The optimal rate of convergence of error for k-nn median regression estimates (Q1083785)

Let \((X,Y),(X_ 1,Y_ 1),...,(X_ n,Y_ n)\) be i.i.d. random vectors where Y is one-dimensional. It is desired to estimate the conditional median \(\xi\) (X) of Y using \(Z_ n=\{X_ i,Y_ i),i=1,...,n\}\) and X. Denote by \(\xi_{nk}(X,Z_ n)\) the k-nn estimate of \(\xi\) (X), and put \[ H_{nk}(Z_ n)=E\{| \xi_{nk}(X,Z_ n)-\xi (X)| | Z_ n\}, \] the conditional mean absolute error. This article establishes the optimal convergence rate of \(P(H_{nk_ n}(Z_ n)\geq \epsilon)\), under fairly general assumptions on (X,Y) and \(k_ n\), which tends to \(\infty\) in some suitable way.