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 µ = EX. Let {v ⁽ⁿ⁾₁ , ..., v ⁽ⁿ⁾ _n } ^∞ _n=1 be vectors of nonnegative random variables (weights), independent of the data sequence {X ₁, ..., X _n } ^∞ _n=1 , and put m _n = Σ ⁿ _i=1 v ⁽ⁿ⁾ _i . Consider v ⁽ⁿ⁾₁ X ₁, ..., v ⁽ⁿ⁾ _n X _n , a bootstrap sample, resulting from re-sampling or stochastically re-weighing the random sample X ₁, ..., X _n , n ≥ 1. Put $\bar X_n = \sum\nolimits_{i = 1}^n {X_i } /n$, the original sample mean, and define $\bar X_{m_n }^* = \sum\nolimits_{i = 1}^n {v_i^{(n)} X_i /m_n }$, where m _n := Σ ⁿ _i=1 v ⁽ⁿ⁾ _i , the bootstrap sample mean. Thus, $\bar X_{m_n }^* - \bar X_n = \sum\nolimits_{i = 1}^n {\left( {v_i^{(n)} /m_n - 1/n} \right)X_i }$. Put V ² _n = Σ ⁿ _i=1 (v ⁽ⁿ⁾ _i /m _n − 1/n)² and let S ² _n , $S_{m_n }^{*2}$ respectively be 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 → ∞, max_1≤i≤n (v ⁽ⁿ⁾ _i /m _n − 1/n)²/V ² _n = o(1) almost surely or in probability on the probability space of the weights. In consequence of these maximum negligibility conditions on the weights, a characterization of the validity of this approach to the bootstrap is obtained as a direct consequence of the Lindeberg-Feller central limit theorem (CLT). This view of justifying the validity of bootstrapping i.i.d. observables 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 → ∞ that differ from requiring m _n /n to be in the interval [λ ₁, λ ₂] with 0 < λ ₁ < λ ₂ < ∞ as in Mason and Shao (2001). The validity of the bootstrapped t-intervals is established for both approaches to conditioning.