We are interested in the estimation of a parameter $$\theta $$ θ that maximizes a certain criterion function depending on an unknown, possibly infinite-dimensional nuisance parameter h . A common estimation procedure consists in maximizing the corresponding empirical criterion, in which the nuisance parameter is replaced by a nonparametric estimator. In the literature, this research topic, commonly referred to as semiparametric M -estimation, has received a lot of attention in the case where the criterion function satisfies certain smoothness properties. In certain applications, these smoothness conditions are, however, not satisfied. The aim of this paper is therefore to extend the existing theory on semiparametric M -estimators, in order to cover non-smooth M -estimators as well. In particular, we develop ‘high-level’ conditions under which the proposed M -estimator is consistent and has an asymptotic limit. We also check these conditions for a specific example of a semiparametric M -estimator coming from the area of classification with missing data.

中文翻译：

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具有非平滑准则函数的半参数 M 估计

我们对参数 $$\theta $$ θ 的估计感兴趣，该参数根据未知的、可能是无限维的有害参数 h 最大化某个标准函数。一个常见的估计程序包括最大化相应的经验标准，其中干扰参数被非参数估计量取代。在文献中，这个研究课题，通常被称为半参数 M 估计，在准则函数满足某些平滑特性的情况下受到了很多关注。然而，在某些应用中，这些平滑条件并不满足。因此，本文的目的是扩展关于半参数 M 估计量的现有理论，以便也涵盖非光滑 M 估计量。特别是，我们开发了“高级”条件，在这些条件下，所提出的 M 估计量是一致的并且具有渐近极限。我们还检查了这些条件，以获取来自具有缺失数据的分类领域的半参数 M 估计量的特定示例。