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On the population least-squares criterion in the monotone single index model

机译：单调单调单索引模型中的人口最小二乘标准

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摘要

Monotone single index models have gained increasing popularity over the past decades due to their flexibility and versatile use in diverse areas. Semi-parametric estimators such as the least squares and maximum likelihood estimators of the unknown index and monotone ridge function were considered to make inference in such models without having to choose some tuning parameter. Description of the asymptotic behavior of those estimators crucially depends on acquiring a good understanding of the optimization problems associated with the corresponding population criteria. In this paper, we give several insights into these criteria by proving existence of minimizers thereof over general classes of parameters. In order to describe these minimizers, we prove different results which give the direction of variation of the population criteria in general and in the special case where the common distribution of the covariates is Gaussian. A complementary simulation study was performed and whose results give support to our main theorems.
机译：由于它们在不同地区的灵活性和多功能使用，单调单位索引模型在过去的几十年里取得了越来越受欢迎。诸如未知索引和单调脊函数的最小二乘和最大似然估计的半参数估计器被认为是在这些模型中推断，而无需选择一些调谐参数。对这些估计器的渐近行为的描述至关重要地取决于获取对与相应人口标准相关的优化问题的良好理解。在本文中，我们通过在一般参数类别中证明其最小化合物的存在，我们对这些标准进行了多次见解。为了描述这些最小化者，我们证明了不同的结果，它一般提供人口标准的变化方向，在协变量的共同分布是高斯的特殊情况下。进行了互补模拟研究，其结果为我们的主要定理提供了支持。

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