Estimation and hypothesis test for partial linear single-index multiplicative models are considered in this paper. To estimate unknown single-index parameter, we propose a profile least product relative error estimator coupled with a leave-one-component-out method. To test a hypothesis on the parametric components, a Wald-type test statistic is proposed. We employ the smoothly clipped absolute deviation penalty to select relevant variables. To study model checking problem, we propose a variant of the integrated conditional moment test statistic by using linear projection weighting function, and we also suggest a bootstrap procedure for calculating critical values. Simulation studies are conducted to demonstrate the performance of the proposed procedure and a real example is analyzed for illustration.

中文翻译：

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偏线性单指数乘法模型的估计和假设检验

本文考虑了部分线性单指数乘法模型的估计和假设检验。为了估计未知的单指标参数，我们提出了一种轮廓最小乘积相对误差估计器和一种留一成分法。为了检验关于参数分量的假设，提出了 Wald 型检验统计量。我们采用平滑剪裁的绝对偏差惩罚来选择相关变量。为了研究模型检查问题，我们提出了一种使用线性投影加权函数的综合条件矩检验统计量的变体，并且我们还提出了用于计算临界值的引导程序。进行了模拟研究以证明所提出的程序的性能，并分析了一个真实的例子以进行说明。