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.

中文翻译：

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单调单指标模型中的总体最小二乘准则

单调单索引模型由于其灵活性和在不同领域的多功能用途，在过去几十年中越来越受欢迎。半参数估计量（例如未知指数和单调脊函数的最小二乘和最大似然估计量）被认为可以在此类模型中进行推理，而无需选择某些调整参数。对这些估计量的渐近行为的描述关键取决于对与相应总体标准相关的优化问题的充分理解。在本文中，我们通过证明它们在一般参数类别上的最小化器的存在，给出了对这些标准的一些见解。为了描述这些最小化器，我们证明了不同的结果，这些结果给出了总体标准的变化方向，在协变量的共同分布是高斯的特殊情况下。进行了补充模拟研究，其结果支持我们的主要定理。