Uniformly consistent bootstrap confidence intervals

Abstract

The bootstrap methods are widely used for constructing confidence intervals.

However, the conventional bootstrap fails to be consistent under some nonstandard

circumstances. The m out of n bootstrap is usually adopted to restore

consistency, provided that a correct convergence rate can be specified for the

plug-in estimators. In this thesis, we re-investigate the asymptotic properties of

the bootstrap in a moving-parameter framework in which the underlying distribution

is allowed to depend on n. We consider the problem of setting uniformly

consistent confidence intervals for two non-regular cases: (1) the smooth function

models with vanishing derivatives; and (2) the M-estimation with non-regular

conditions.

Under the moving-parameter setup, neither the conventional bootstrap nor

the m out of n bootstrap is shown uniformly consistent over the whole parameter space. The results reflect to some extent finite-sample anomalies that cannot be

explained by conventional, fixed-parameter, asymptotics. We propose a weighted

bootstrap procedure for constructing uniformly consistent bootstrap confidence

intervals, which does not require explicit specification of the convergence rate

of the plug-in estimator. Under the smooth function models, we also propose

a modified n out of n bootstrap procedure in special cases where the smooth

function is applied to estimators that are uniformly bootstrappable. The estimating

function bootstrap is also successfully employed for the latter model

and enjoys computational advantages over the weighted bootstrap. We illustrate

our findings by comparing the finite-sample coverage performances of the different

bootstrap procedures. The stable performance of the proposed methods,

contrasts sharply with the erratic coverages of the n out of n and m out of n

bootstrap intervals, a result in agreement with our theoretical findings.