Bootstrapping extremes of random variables
Bootstrap, introduced by Efron (1979), is a general method of estimating the distribution Gn of a function R(Xn,F) of the data Xn = X1,X2,...,Xn and the underlying distribution F. It consists of generating Xn,m* = X1*,X2*,...,Xm* by resampling the data Xn and studying the conditional distribution Hn,m of the function R(Xn,m*, Fn) of Xn,m* and the empirical distribution Fn. Bootstrap is said to be consistent if the difference of Gn and Hn,m converges to zero in probability (or with probability 1) as the sample size n and resample size m tend to infinity. This thesis studies the consistency of Efron's bootstrap and the moving block bootstrap (MBB) for extremes of both independent and dependent random variables;After an introduction and a literature review in Chapter 1, the case of independently and identically distributed random variables is studied in Chapter 2. It is shown that Efron's bootstrap is consistent if and only if mn-1 → 0;In Chapter 3, the case of a stationary sequence of random variables is studied. It is shown that both Efron's bootstrap and the MBB is consistent if mn-1 → 0 and conditions D(un) and D'(un) of Leadbetter (1974) hold. It is also shown that the MBB is consistent even without D'(un), and thus it works for wider class of stationary processes than Efron's bootstrap does;In Chapter 4, we propose a method of constructing confidence intervals for the lower endpoint of a cdf by applying the smoothed bootstrap to Weissman's (1981) Statistics and Probability; The method is extended to the case of the type II censoring. Some Monte Carlo simulation results are included.