标题： 对学生t统计量自助法的再思考 显示英文标题

标题： Another look at Bootstrapping the Student t-statistic

摘要： 设 X, X_1,X_2,... 是一列独立同分布的随机变量，均值为$\mu=E X$。 设${v_1^{(n)},...,v_n^{(n)}}_{n=1}^\infty$为非负随机变量（权重）的向量，独立于数据序列${X_1,...,X_n}_{n=1}^\infty$，并令$m_n=\sumn v_i^{(n)}$。 考虑$ X^{*}_1,..., X^{*}_{m_n}$，$m_n\geq 1$，一个自助样本，由对随机样本$X_1,...,X_n$，$n\geq 1$进行重采样或随机加权得到。 设$\bar{X}_n= \sumn X_i/n$，原始样本均值，并定义 $\bar{X^*}_{m_n}=\sumn v_i^{(n)} X_i/m_n$，自助样本均值。 因此， $\bar{X^*}_{m_n}- \bar{X}_n=\sumn ({v_i^{(n)}}/{m_n}-{1}/{n}) X_i$。 将$V_n^{2}=\sumn ({v_i^{(n)}}/{m_n}-{1}/{n})^2$设为，并让$S_n^{2}$，$S_{m_{n}}^{*^{2}}$分别为原始样本方差和引导样本方差。 本文的主要目的是研究在假设当$n,m_n\to \infty$时，通过条件化权重来研究自举的$t$-统计量$T_{m_n}^{*}:= (\bar{X^*}_{m_n}- \bar{X}_n)/(S_n V_n)$和$T_{m_n}^{**}:= \sqrt{m_n}(\bar{X^*}_{m_n}- \bar{X}_n)/ S_{m_{n}}^{*} $的渐近行为，$\max_{1\leq i \leq n}({v_i^{(n)}}/{m_n}-{1}/{n})^2\big/ V_n^{2}=o(1)$在权重的概率空间上几乎必然或以概率成立。这种证明自举有效性的观点被认为是一种新的方法。当通过重采样探索随机样本中包含的信息性质时，实践中自然会需要这种方法。 在数据条件下，也重新审视了Efron的bootstrap权重，其条件为$n,m_n$，其中$n\to \infty $与要求$m_n /n$位于区间$(\lambda_1,\lambda_2)$内不同，且0<\lambda _1 <\lambda _2 <\infty ，如Mason和Shao所述。 此外，对于条件处理的两种方法，引导的$t$-区间的有效性得到了证明。 显示英文摘要

摘要： Let X, X_1,X_2,... be a sequence of i.i.d. random variables with mean $\mu=E X$. Let ${v_1^{(n)},...,v_n^{(n)}}_{n=1}^\infty$ be vectors of non-negative random variables (weights), independent of the data sequence ${X_1,...,X_n}_{n=1}^\infty$, and put $m_n=\sumn v_i^{(n)}$. Consider $ X^{*}_1,..., X^{*}_{m_n}$, $m_n\geq 1$, a bootstrap sample, resulting from re-sampling or stochastically re-weighing a random sample $X_1,...,X_n$, $n\geq 1$. Put $\bar{X}_n= \sumn X_i/n$, the original sample mean, and define $\bar{X^*}_{m_n}=\sumn v_i^{(n)} X_i/m_n$, the bootstrap sample mean. Thus, $\bar{X^*}_{m_n}- \bar{X}_n=\sumn ({v_i^{(n)}}/{m_n}-{1}/{n}) X_i$. Put $V_n^{2}=\sumn ({v_i^{(n)}}/{m_n}-{1}/{n})^2$ and let $S_n^{2}$, $S_{m_{n}}^{*^{2}}$ respectively be the the original sample variance and the bootstrap sample variance. The main aim of this exposition is to study the asymptotic behavior of the bootstrapped $t$-statistics $T_{m_n}^{*}:= (\bar{X^*}_{m_n}- \bar{X}_n)/(S_n V_n)$ and $T_{m_n}^{**}:= \sqrt{m_n}(\bar{X^*}_{m_n}- \bar{X}_n)/ S_{m_{n}}^{*} $ in terms of conditioning on the weights via assuming that, as $n,m_n\to \infty$, $\max_{1\leq i \leq n}({v_i^{(n)}}/{m_n}-{1}/{n})^2\big/ V_n^{2}=o(1)$ almost surely or in probability on the probability space of the weights. This view of justifying the validity of the bootstrap is believed to be new. The need for it arises naturally in practice when exploring the nature of information contained in a random sample via re-sampling, for example. Conditioning on the data is also revisited for Efron's bootstrap weights under conditions on $n,m_n$ as $n\to \infty $ that differ from requiring $m_n /n$ to be in the interval $(\lambda_1,\lambda_2)$ with 0< \lambda_1 < \lambda_2 < \infty as in Mason and Shao. Also, the validity of the bootstrapped $t$-intervals for both approaches to conditioning is established.