随机加权法的渐近展开

THE EDGEWORTH EXPANSION FOR THE RANDOM WEIGHTING METHOD

摘要: 本文讨论了样本均值规范化误差 (\barX-\mu) / \sigma 之分布函数 F_n(x) 的模拟问题. 在本文中采用了一种与 Bootstrap 方法不同的方法——随机加权法。记 F_n^* 为 \boldsymbol\Sigma\left(\barX_i-\barX\right) V_i / \sigma^*\left(\boldsymbol\Sigma\left(X_i-\barX\right) V_i\right) 之分布，其中 \left(V_1, \cdots, V_n\right) 遵从 Dirichlet 分布， \sigma^* 2 表示 X_1, \cdots, X_n 固定之下 \Sigma\left(X_i-\barX\right) V_i 之方差。本文的主要结论是当 E\left|X_i\right|^3< \infty 时, \sqrtn \sup _x\left|F_n^*(x)-F_n(x)\right| \rightarrow 0, a.e.

Abstract: 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.)