Reduced bootstrap for the median

Abstract

In this paper we study a modified bootstrap that consists of only considering those bootstrap samples satisfying k1 ≤ νn ≤ k2, for some 1 ≤ k1 ≤ k2 ≤ n, where νn is the number of distinct original observations in the bootstrap sample. We call it reduced bootstrap, since it only uses a portion of the set of all possible bootstrap samples. We show that, under some conditions on k1 and k2, the reduced bootstrap consistently estimates the distribution and the variance of the sample median. Unlike the ordinary bootstrap, the reduced bootstrap variance estimator does not require conditions on the population generating the data to be a consistent estimator, but does rely an adequate choice of k1 and k2. Since several choices of k1 and k2 yield consistent estimators, we compare the finite sample performance of the corresponding estimators through a simulation study. The simulation study also considers consistent variance estimators proposed by other authors.