THE EDGEWORTH EXPANSION FOR THE RANDOM WEIGHTING METHOD

The random weighting method, which differs from bootstrap, is a new approch to estimate of the distribution of pivotal statistics. In this paper, we develop an Edgeworth expansion for the random weighting distribution of sample mean. Let F_n(x) be distribution function of (\barX-\mu) / \sigma and F_n^*(x) distribution function of \Sigma\left(X_1-\barX\right) V_d / \sigma^*\left(\Sigma\left(X_1-\barX\right) V_i\right), where the distribution of \left(v_1, v_2, \cdots, v_n\right) is a Dirichlet D(4,4, \cdots, 4) distribution and \sigma^* 2 the variance of \left(\Sigma\left(X_i-\barX\right) v_i\right) given X_1, X_2, \cdots, X_n. Using the expansion we have: If E|X|^3< \infty, then \sqrtn \sup _\epsilon\left|F_n^*(x)-F_n(x)\right| \rightarrow 0 \quad (a.e.)