We propose a unified framework for establishing existence of nonparametric M-estimators, computing the corresponding estimates, and proving their strong consistency when the class of functions is exceptionally rich. In particular, the framework addresses situations where the class of functions is complex involving information and assumptions about shape, pointwise bounds, location of modes, height at modes, location of level-sets, values of moments, size of subgradients, continuity, distance to a "prior" function, multivariate total positivity, and any combination of the above. The class might be engineered to perform well in a specific setting even in the presence of little data. The framework views the class of functions as a subset of a particular metric space of upper semicontinuous functions under the Attouch-Wets distance. In addition to allowing a systematic treatment of numerous M-estimators, the framework yields consistency of plug-in estimators of modes of densities, maximizers of regression functions, level-sets of classifiers, and related quantities, and also enables computation by means of approximating parametric classes. We establish consistency through a one-sided law of large numbers, here extended to sieves, that relaxes assumptions of uniform laws, while ensuring global approximations even under model misspecification.

中文翻译：

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受约束的 M 估计量的变分分析

我们提出了一个统一的框架，用于建立非参数 M 估计量的存在性，计算相应的估计量，并在函数类异常丰富时证明它们的强一致性。特别是，该框架解决了函数类复杂的情况，涉及关于形状、逐点边界、模式位置、模式高度、水平集位置、矩值、次梯度大小、连续性、到一个“先验”函数、多元总正性，以及上述的任意组合。即使在数据很少的情况下，该类也可能被设计为在特定设置中表现良好。该框架将函数类视为 Attouch-Wets 距离下的上半连续函数的特定度量空间的子集。除了允许对众多 M 估计器进行系统处理之外，该框架还产生了密度模式的插件估计器、回归函数的最大化器、分类器的水平集和相关量的一致性，并且还能够通过近似计算参数类。我们通过单边大数定律建立一致性，这里扩展到筛子，放宽了统一定律的假设，同时即使在模型错误指定的情况下也确保全局近似。