BOOTSTRAP METHOD AND EMPIRICAL PROCESS

MASAFUMI AKAHIRA^{1} AND KEI TAKEUCHI^{2}

1Institute of Mathematics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan 2Research Center for Advanced Science and Technology, University of Tokyo,

4-6-1 Komaba, Meguro-ku, Tokyo 156, Japan

(Received August 24, 1989; revised February 22, 1990)

Abstract. In this paper we consider the sampling properties of the bootstrap process, that is, the empirical process obtained from a random sample of size n (with replacement) of a fixed sample of size n of a continuous distribution. The cumulants of the bootstrap process are given up to the order n^{−1} and their unbiased estimation is discussed. Furthermore, it is shown that the bootstrap process has an asymptotic minimax property for some class of distributions up to the order n^{−1/2}.

Key words and phrases: Bootstrap process, cumulants, unbiased estimators, asymptotic minimax property.