Extensions of the Markov chain marginal bootstrap

The Markov chain marginal bootstrap (MCMB) was introduced by He and Hu [2002. Markov chain marginal bootstrap. J. Amer. Statist. Assoc. 97(459) (2002) 783-795] as a bootstrap-based method for constructing confidence intervals or regions for a wide class of M-estimators in linear regression and maximum likelihood estimators in certain parametric models. In this article we discuss more general applications of MCMB-A, an extension of the MCMB algorithm, which was first proposed in Kocherginsky et al. [2005. Practical confidence intervals for regression quantiles. J. Comput. Graphical Statist. 14, 41-55] for quantile regression models. We also present a further extension of the MCMB algorithm, the B-transformation, which is a transformation of the estimating equations, aiming to broaden the applicability of the MCMB algorithm to general estimating equations that are not necessarily likelihood-based. We show that applying the A- and B-transformations jointly enables the MCMB algorithm to be used for inference related to a very general class of estimating equations. We illustrate the use of the MCMB-AB algorithm with a nonlinear regression model with heteroscedastic error distribution.