It has been known for a long time that for bootstrapping the probability distribution of the maximum of a sample consistently, the bootstrap sample size needs to be of smaller order than the original sample size. See Jun Shao and Dongsheng Tu (1995), Ex. 3.9,p. 123. We show that the same is true if we use the bootstrap for estimating an intermediate quantile.

,
hdl.handle.net/1765/541
Econometric Institute Research Papers
Erasmus School of Economics

Geluk, J.L, & de Haan, L.F.M. (2002). On bootstrap sample size in extreme value theory (No. EI 2002-40). Econometric Institute Research Papers. Retrieved from http://hdl.handle.net/1765/541