Let (X,Y) be a Rd R-valued random vector. In regression analysis one wants to estimate the regression function m(x):=E(Y|X=x) from a data set. In this paper we consider the convergence rate of the error for the k nearest neighbor estimators in case that m is (p,C)-smooth. It is known that the minimax rate is unachievable by any k nearest neighbor estimator for p > 1.5 and d=1. We generalize this result to any d ≥ 1. Throughout this paper, we assume that the data is independent and identically distributed and as an error criterion we use the expected L2 error.

In this paper we consider the two-class classification problem with high-dimensional data. It is important to find a class of distributions such that we cannot expect good performance in classification for any classifier. In this paper, when two population variance-covariance matrices are different, we give a reasonable sufficient condition for distributions such that the misclassification rate converges to the worst value as the dimension of data tends to infinity for any classifier. Our results can give guidelines to decide whether or not an experiment is worth performing in many fields such as bioinformatics.