With the advancement in modern computing facilities, the use of more complicated/intricate statistical procedures are becoming increasingly popular, and usually the complex characterizations of the estimators involved lead to non-standard asymptotic behavior (i.e., estimators converge at rates different from the usual square-root n rate in standard parametric problems and/or have non-Gaussian limit distributions). Developing valid inferential tools (e.g., construction of confidence intervals (CIs), hypothesis tests) in this setup is indeed very challenging. The bootstrap is certainly the most popular inferential tool employed in such problems. But a common feature in a majority of these problems is that the usual “with replacement” bootstrap is not consistent. I have explored bootstrap based inference in non-standard problems in the papers [1, 2, 3, 4].

These papers consider different non-standard problems, characterized by different rates of convergence of the normalized estimator. Although the m-out-of-n bootstrap or subsampling can be used as an alternative to construct CIs in these situations, it is usually observed that a careful and explicit use of the model assumptions can lead to bootstrap procedures that are consistent and have better finite sample performance. Often such model-based bootstrap methods are completely automated (see e.g., [2]), and thus the choice of tuning parameter(s), like the block size (in subsampling and the m-out-of-n bootstrap), can be avoided.

The following papers cover a broad spectrum of non-standard problems: the asymptotic distribution of the estimators vary from normal distributions to complex functionals of Gaussian processes to minimizers of compound Poisson processes (with drifts).

References

1. 1.Sen, B., Banerjee, M., and Woodroofe, M. (2010). Inconsistency of Bootstrap: the Grenander Estimator. Ann. Statist., 38, 1953-1977.
   

2. 2.McKeague, I. and Sen, B. (2010). Fractals with point impact in functional linear regression. Ann. Statist., 38, 2559-2586.
   

3. 3.Seijo, E. and Sen, B. (2011). Change point in stochastic design regression and the bootstrap. Ann. Statist., 39, 1580-1607.
   

4. 4.Sen, B. and Woodroofe, M. (2011). Bootstrap Confidence Intervals for Isotonic Estimators in a Stereological Problem. Bernoulli (to appear).