By optimizing index functions against different outcomes, we propose a multivariate single-index model (SIM) for development of medical indices that simultaneously work with multiple outcomes. Fitting of a multivariate SIM is not fundamentally different from fitting a univariate SIM, as the former can be written as a sum of multiple univariate SIMs with appropriate indicator functions. What have not been carefully studied are the theoretical properties of the parameter estimators. Because of the lack of asymptotic results, no formal inference procedure has been made available for multivariate SIMs. In this paper, we examine the asymptotic properties of the multivariate SIM parameter estimators. We show that, under mild regularity conditions, estimators for the multivariate SIM parameters are indeed n-consistent and asymptotically normal. We conduct a simulation study to investigate the finite-sample performance of the corresponding estimation and inference procedures. To illustrate its use in practice, we construct an index measure of urine electrolyte markers for assessing the risk of hypertension in individual subjects.

中文翻译：

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多元单指标模型中的大样本估计和推理

通过针对不同结果优化指标函数，我们提出了一种多元单指标模型 (SIM)，用于开发同时处理多种结果的医学指标。拟合多变量 SIM 与拟合单变量 SIM 没有根本区别，因为前者可以写成具有适当指示函数的多个单变量 SIM 的总和。尚未仔细研究的是参数估计器的理论特性。由于缺乏渐近结果，还没有正式的推理过程可用于多变量 SIM。在本文中，我们研究了多元 SIM 参数估计器的渐近特性。我们表明，在温和的规律性条件下，多元 SIM 参数的估计量确实是 n 一致且渐近正态的。我们进行了一项模拟研究，以研究相应估计和推理程序的有限样本性能。为了说明其在实践中的应用，我们构建了尿电解质标志物的指数测量，用于评估个体受试者的高血压风险。