2012/09/18 by Miklós Csörgő, Csorgo, Miklos, Yuliya V. Martsynyuk +3 · 1 citation
Computer Science · Decision Sciences · Mathematics · #60K35 #62F40 #Bayesian Methods and Mixture Models #FOS: Mathematics #Mathematical Approximation and Integration #Probability and Risk Models #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.1209.4089
openalex publication_date 2012/09/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X, X1,X2,... be a sequence of i.i.d. random variables with mean μ=E X. Let v1(n),...,vn(n)n=1^∞ be vectors of non-negative random variables (weights), independent of the data sequence X1,...,Xnn=1^∞, and put mn=\sumn vi(n). Consider X*1,..., X*mn, mn≥ 1, a bootstrap sample, resulting from re-sampling or stochastically re-weighing a random sample X1,...,Xn, n≥ 1. Put Xn= \sumn Xi/n, the original sample mean, and define X^*mn=\sumn vi(n) Xi/mn, the bootstrap sample mean. Thus, X^*mn- Xn=\sumn (vi(n)/mn-1/n) Xi. Put Vn2=\sumn (vi(n)/mn-1/n)2 and let Sn2, S_mn^*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 Tmn*:= (X^*mn- Xn)/(Sn Vn) and Tmn**:= √(mn)(X^*mn- Xn)/ S_mn* in terms of conditioning on the weights via assuming that, as n,mn→ ∞, max1≤ i ≤ n(vi(n)/mn-1/n)2/ Vn2=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,mn as n→ ∞ that differ from requiring mn /n to be in the interval (λ1,λ2) with 0< λ1 < λ2 < ∞ as in Mason and Shao. Also, the validity of the bootstrapped t-intervals for both approaches to conditioning is established.