摘要:
本文讨论了样本均值规范化误差 $(\bar{X}-\mu) / \sigma$ 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 $\boldsymbol{\Sigma}\left(\bar{X}_i-\bar{X}\right) V_i / \sigma^*\left(\boldsymbol{\Sigma}\left(X_i-\bar{X}\right) V_i\right)$ 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 $\Sigma\left(X_i-\bar{X}\right) V_i$ 之方差。本文的主要结论是当 $E\left|X_i\right|^3< \infty$ 时, $\sqrt{n} \sup _x\left|F_n^*(x)-F_n(x)\right| \rightarrow 0$, 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 $(\bar{X}-\mu) / \sigma$ and $F_n^*(x)$ distribution function of $\Sigma\left(X_1-\bar{X}\right) V_d /$ $\sigma^*\left(\Sigma\left(X_1-\bar{X}\right) V_i\right)$, 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 $\left(\Sigma\left(X_i-\bar{X}\right) v_i\right)$ given $X_1, X_2, \cdots, X_n$. Using the expansion we have: If $E|X|^3< \infty$, then $\sqrt{n} \sup _\epsilon\left|F_n^*(x)-F_n(x)\right| \rightarrow 0 \quad$ (a.e.)
下载:
