Generalized Bootstrap for Estimating Equations

We introduce a generalized bootstrap technique for estimators obtained by solving estimating equations. Some special cases of this generalized bootstrap are the classical bootstrap of Efron, the deleted jackknife and variations of the Bayesian bootstrap. The use of the proposed technique is discussed in some examples. Distributional consistency of the method is established and an asymptotic representation of the resampling variance estimator is obtained. 1. Introduction. One of the most popular ways of obtaining estimators for parameters in statistics is by solving " estimating equations. " Examples are abundant in the contexts of quasi-likelihood methods, time series, bio-statistics, stochastic processes, spatial statistics, robust inference, survey sampling and other areas. Godambe (1991) and Basawa, Godambe and Tay-lor (1997) contain extensive discussions on estimating equations. In this paper we introduce a generalized bootstrap technique for estimators obtained by solving estimating equations. We use the following framework: Suppose {φ ni (Z ni , β), 1 ≤ i ≤ n, n ≥ 1} is a triangular sequence of functions taking values in R p , {Z ni } being a sequence of observable random variables and β ∈ B ⊂ R p. Assume that Eφ ni (Z ni , β 0) = 0, 1 ≤ i ≤ n, n ≥ 1 for some unique β 0 ∈ B. The " parameter " β 0 is unknown, and its estimatorˆβ n is obtained by solving (often uniquely) the estimating equations