Bootstrapping Student t统计量
Another look at Bootstrapping the Student t-statistic
摘要：
设X，X_1，X_2，.是一个i.i.d.序列。随机变量与平均$μ=E X$。令${v_1^{(n)},...,v_n^{(n)}}_{n=1}^\infty$为非负随机变量（权重）的向量，独立于数据序列${X_1,...,X_n}_{n=1}^\infty$，并将$m_n=\sumn v_i^{(n)}$。考虑$ X^{*}_1,..., X^{*}_{m_n}$、$m_n\geq 1$，这是一种自助样本，由随机样本$X_1,...,X_n$、$n\geq 1$的重新采样或随机重新称重产生。输入原始样本均值$\bar{X}_n= \sumn X_i/n$，并定义自举样本均值$\bar{X^*}_{m_n}=\sumn v_i^{(n)} X_i/m_n$。因此，$\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$和$S_n^{2}$、$S_{m_{n}}^{*^{2}}$分别作为原始样本方差和自助样本方差。本文的主要目的是研究自举$t$ -统计$T_{m_n}^{*}:= (\bar{X^*}_{m_n}- \bar{X}_n)/(S_n V_n)$和$T_{m_n}^{**}:= \sqrt{m_n}(\bar{X^*}_{m_n}- \bar{X}_n)/ S_{m_{n}}^{*} $在权重条件下的渐近行为，假设$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)$几乎必然或概率在权重的概率空间上。这种证明自举有效性的观点被认为是新的。例如，在实践中，当通过重新采样探索随机样本中包含的信息的性质时，自然会出现对它的需求。在$n,m_n$作为$n\to \infty $的条件下，也重新审视了Efron的自举权重的数据条件，该条件不同于Mason和Shao中要求$m_n /n$在0< λ_1 < λ_2 < \infty的$(λ_1,λ_2)$区间内。此外，自举$t$的有效性-间隔的两种方法来调节建立。
英文摘要：
Let X, X_1,X_2,... be a sequence of i.i.d. random variables with mean $μ=E X$. 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 $(λ_1,λ_2)$ with 0< λ_1 < λ_2 < \infty as in Mason and Shao. Also, the validity of the bootstrapped $t$-intervals for both approaches to conditioning is established.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
