Statistica Sinica 14(2004), 1179-1198

REDUCED BOOTSTRAP FOR THE MEDIAN

M. D. Jiménez-Gamero, J. Muñoz-García and R. Pino-Mejías

Universidad de Sevilla

Abstract: In this paper we study a modified bootstrap that consists of only considering those bootstrap samples satisfying $k_1 \leq \nu_n \leq k_2$, for some $1\leq k_1 \leq k_2 \leq n$, where $\nu_n$ 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 $k_1$ and $k_2$, 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 $k_1$ and $k_2$. Since several choices of $k_1$ and $k_2$ 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.

Key words and phrases: Bootstrap, consistency, distribution estimation, sample median, variance estimation.