Another look at Bootstrapping the Student t-statistic

Abstract

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.