This paper develops distribution theory and bootstrap-based inference methods for a broad class of convex pairwise difference estimators. These estimators minimize a kernel-weighted convex-in-parameter function over observation pairs with similar covariates, where the similarity is governed by a localization (bandwidth) parameter. While classical results establish asymptotic normality under restrictive bandwidth conditions, we show that valid Gaussian and bootstrap-based inference remains possible under substantially weaker assumptions. First, we extend the theory of small bandwidth asymptotics to convex pairwise difference estimation settings, deriving robust Gaussian approximations even when a smaller than standard bandwidth is used. Second, we employ a debiasing procedure based on generalized jackknifing to enable inference with larger bandwidths, while preserving convexity of the objective function. Third, we construct a novel bootstrap method that adjusts for bandwidth-induced variance distortions, yielding valid inference across a wide range of bandwidth choices. Our proposed inference method enjoys demonstrably greater robustness, while retaining the practical appeal of convex pairwise difference estimators.

翻译：本文针对一大类凸成对差分估计量发展了分布理论与基于自助法的推断方法。这类估计量通过核加权的方式最小化参数中的凸函数，该凸函数基于协变量相似的观测对进行构造，其中相似性由局部化（带宽）参数控制。经典结果在严格的带宽条件下建立了渐近正态性，而本文证明在显著更弱的假设下，有效的高斯近似及自助法推断仍可实现。首先，我们将小带宽渐近理论拓展至凸成对差分估计框架，即使在带宽小于标准值的情况下也能推导出稳健的高斯近似。其次，采用基于广义刀切法的去偏程序，在保持目标函数凸性的同时，支持使用更大带宽进行推断。第三，构建了一种新型自助法，该方法可校正由带宽引起的方差畸变，从而在广泛的带宽选择范围内实现有效推断。本文提出的推断方法在保留凸成对差分估计量实用优势的同时，展现出更强的稳健性。