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Abstract:
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In this talk, I will discuss finite sample approximations to the supremum of a non-degenerate U-process of a general order indexed by a function class. We are primarily interested in situations where the function class as well as the underlying distribution change with the sample size, and the U-process itself is not weakly convergent as a process. We first consider Gaussian approximations and derive coupling and Kolmogorov distance bounds. Such Gaussian approximations are, however, not often directly usable in statistical problems since the covariance function of the approximating Gaussian process is unknown. This motivates us to study bootstrap-type approximations to the U-process supremum. We propose a novel jackknife multiplier bootstrap (JMB) tailored to the U-process, and derive coupling and Kolmogorov distance bounds for the proposed JMB method. We also discuss applications of the general approximation results to testing for qualitative features of nonparametric functions based on generalized local U-processes. This talk is based on joint work with Xiaohui Chen (UIUC).
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