Bootstrapping U -statistics: applications in least squares and robust regression
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Bootstrapping U-statistics: applications in least squares and robust regression Wenyu Jiang Queen’s University, Ontario, Canada
John D. Kalbfleisch University of Michigan, Ann Arbor, USA
Abstract Suppose that inference about parameters of interest is to be based on an unbiased estimating function that is U-statistic of degree 1 or 2. We define pivotal or approximate pivotal statistics based on such estimating functions and propose asymptotic approximations as well as estimating function bootstrap (EFB) methods based on resampling the estimated terms in the estimating functions. These methods are justified asymptotically and lead to confidence intervals produced directly from the estimating function based quantities that are pivotal or asymptotically pivotal. Particular examples in this class of estimating functions arise in least-squares regression, robust La estimation (1 ≤ a < 2), Wilcoxon rank regression and other related estimation problems. The proposed methods are evaluated in simulations and applied to a real data set. When compared with existing asymptotic and resampling methods, the EFB method is found to be more accurate in small sample situations and robust to violation of model assumptions. AMS (2000) subject classification. Primary 62G09; Secondary 62G08. Keywords and phrases. Bootstrap, estimating functions, La estimation, leastsquares regression, pivotal, resampling methods, studentization, U-statistics, Wilcoxon rank regression.
1 Introduction In the literature, a U-statistic is typically real-valued and used as an estimator of an unknown quantity such as a moment, a quantile or a correlation (Serfling, 1980). In this paper, we consider estimating functions with U-statistic structure, and propose inference procedures for the parameters of interest. Let Z1 , . . . , Zn be independent and identically distributed random vectors. Let θ ∈ Rp be a vector of parameters and suppose that estimation
Bootstrapping U-statistics of θ is based on an unbiased estimating function −1 n S(θ) = h(Zi1 , . . . , ZiK ; θ) K
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(1.1)
1≤i1
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