抄録
Independent observations X1, X2, …, Xn are made on a distribution F on Rd. To devide these observations into k clusters, first choose a vector of optimal cluster centers bn=(bn1, bn2, …, bnk) to minimize Wn(a)=1/nΣni=1min1≤j≤k||Xi-aj||2 as a function of a=(a1, a2, …, ak), then assign each observation to its nearest cluster center. Each bnj is the mean of observations in its cluster. Pollard (1982) obtained a central limit theorem for the means of the k-clusters. In this paper, it is shown that the bootstrap distribution of the centered bn has the same limiting distribution; the argument rests on asymptotic behavior of empirical processes on Vapnik-Chervonenkis classes in triangular array setting. Advantages of the bootstrap methods are discussed and the performance of bootstrap confidence sets is compared with Pollard's confidence sets by Monte Carlo simulation. 2
