On Efron's method II with identification of outlier bootstrap samples
Let \(X=(X_1,\dots,X_n)\) be a sample from an unknown distribution and let \(T(X)\) be some statistic. If \(X^{*1},\dots,X^{*B}\) are some bootstrap resamples from \(X\), \(T^{*i}=T(X^{*i})\), then the bootstrap estimators of the bias and variance of \(T\) are \[ \widehat{bias}_B=B^{-1}\sum_{i=1}^B T^{*i}-T(X),\quad \widehat{var}_B=(B-1)^{-1}\sum_{i=1}^B(T^{*i}-T(X))^2. \] If \(T\) is symmetrically defined in its arguments then (for a fixed \(X\)) \(T^*=T(N_1,N_2,\dots,N_n)\), where \(N_i=card \{X_j^*=X_j\}\). In the naive bootstrap the vectors \(\vec N=(N_1,\dots,N_n)\) are generated from the multinomial distribution \(Mult(n;1/n,\dots,1/n)\). The authors propose a new method of bootstrap resampling (OBS) where the vectors \(\vec N\) are taken from the conditional distribution \(\left. Mult(n;1/n,\dots,1/n)\right| _{Y(\vec N)>C}\), where \(Y(\vec N)\) is the number of different elements in the bootstrapped sample, i.e., the outlier samples with too low number of different elements are excluded from the analysis. For the quadratic statistics \(T^*\), i.e. the statistics of the form \[ T^*=\mu+n^{-1}\sum_{i=1}^n\alpha_i N_i + n^{-2}\{2^{-1}\sum_{i=1}^n\beta_{ii}N_i(N_i-1) + \sum_{i<j}\beta_{ij}N_iN_j\}, \] (\(\alpha_i\), \(\beta_{ij}\) may depend on \(X\)), the mean and variance of OBS bias estimators are computed. Analogous results are obtained for linear \(T^*\) (\(\beta_{ij}=0\)) and OBS variance estimators. It is shown that under some conditions they are more efficient then the naive bootstrap estimators.
scientific article; zbMATH DE number 1336631