This paper discusses a semiparametric single-index model. The link function is allowed to be unbounded and has unbounded support that fill the gap in the literature. The link function is treated as a point in an infinitely many dimensional function space which enables us to derive the estimates for the index parameter and the link function simultaneously. This approach is different from the profile method commonly used in the literature. The estimator is derived from an optimization with the constraint of an identification condition for the index parameter, which solves an important problem in the literature of single-index models. In addition, making use of a property of Hermite orthogonal polynomials, an explicit estimator for the index parameter is obtained. Asymptotic properties of the two estimators of the index parameter are established. Their efficiency is discussed in some special cases as well. The finite sample properties of the two estimators are demonstrated through an extensive Monte Carlo study and an empirical example.

中文翻译：

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约束条件下单指标模型的序列估计

本文讨论了一个半参数单指标模型。允许链接函数是无界的，并且具有填补文献空白的无界支持。链接函数被视为无限多维函数空间中的一个点，这使我们能够同时推导出指数参数和链接函数的估计值。这种方法不同于文献中常用的剖面法。估计量是从具有指标参数识别条件约束的优化中推导出来的，它解决了单指标模型文献中的一个重要问题。此外，利用Hermite正交多项式的性质，得到了指标参数的显式估计量。建立了指标参数的两个估计量的渐近性质。在一些特殊情况下也讨论了它们的效率。这两个估计量的有限样本特性通过广泛的 Monte Carlo 研究和一个经验示例来证明。