ONVERGENCE ON THE CONDITIONAL RISK ESTIMATE IN THE NEAREST NEIGHBOR PREDICTION

Let （X, θ） be a random vector, X∈R^d, θ∈R~1, （X_i, θ_i） be iid random samples of （X, θ）,i=1,…, n, and L_n be the conditional risk in the nearest neighbor （NN） predic- tion under square loss. The estimate of L_n is defined as \hatL_n=\frac1n \sum_j=1^n\left(\theta_j-\theta_n j\right)^2, where θ_nj denotes the NN prediction of θ_j, based on the training sample （X₁, θ₁）, …, (X_j-1, θ_j-1), (X_j+1, θ_j+1), …, （X_n, θ_n） and the observed X_j. Ammusing only the second moment of θ is finite and some other conditions, oonvergenee in series form and in mean is discussed.