In this paper, our goal is to estimate the homogeneous parameter and cluster the heterogeneous parameters in a partially heterogeneous single index model (PHSIM). To achieve the goal, the minimization criterion for such a single index model is first transformed into a least-squares optimization problem in the population form. Based on the least-squares objective function, we introduce an empirical version for the PHSIM. By minimizing such an empirical version, we estimate the homogeneous parameter and the subgroup-averages of the heterogeneous index directions, and then use a fusion penalized method to identify the subgroup structure of the PHSIM. By the proposed methodologies, the homogeneous parameter and the heterogeneous index directions can be consistently estimated, and the heterogeneous parameters can be consistently clustered. Moreover, the new clustering procedure is simple and robust. Simulation studies are carried out to examine the performance of the proposed methodologies.

中文翻译：

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部分异构单索引模型的估计与聚类

在本文中，我们的目标是在部分异构单索引模型（PHSIM）中估计同构参数并将异构参数聚类。为了达到目标，这种单指标模型的最小化标准首先转化为总体形式的最小二乘优化问题。基于最小二乘目标函数，我们为 PHSIM 引入了一个经验版本。通过最小化这样的经验版本，我们估计异构索引方向的同构参数和子组平均值，然后使用融合惩罚方法来识别 PHSIM 的子组结构。通过所提出的方法，可以一致地估计同质参数和异质索引方向，并且可以一致地聚类异质参数。而且，新的聚类过程简单而稳健。进行模拟研究以检查所提出方法的性能。