Zhang Xinyu, Wang Haiying, Ma Yanyuan, Carroll Raymond J
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China,
Department of Mathematics and Statistics, University of New Hampshire, Durham, NH 03824,
J Am Stat Assoc. 2017;112(520):1553-1561. doi: 10.1080/01621459.2016.1219262. Epub 2017 Jun 29.
Prediction precision is arguably the most relevant criterion of a model in practice and is often a sought after property. A common difficulty with covariates measured with errors is the impossibility of performing prediction evaluation on the data even if a model is completely given without any unknown parameters. We bypass this inherent difficulty by using special properties on moment relations in linear regression models with measurement errors. The end product is a model selection procedure that achieves the same optimality properties that are achieved in classical linear regression models without covariate measurement error. Asymptotically, the procedure selects the model with the minimum prediction error in general, and selects the smallest correct model if the regression relation is indeed linear. Our model selection procedure is useful in prediction when future covariates without measurement error become available, e.g., due to improved technology or better management and design of data collection procedures.
预测精度可以说是实践中模型最相关的标准，并且常常是人们所追求的属性。对于存在测量误差的协变量，一个常见的困难是，即使模型完全给定且没有任何未知参数，也无法对数据进行预测评估。我们通过利用具有测量误差的线性回归模型中矩关系的特殊性质来绕过这一固有困难。最终产物是一种模型选择程序，它能实现与没有协变量测量误差的经典线性回归模型相同的最优性质。渐近地，该程序通常会选择具有最小预测误差的模型，如果回归关系确实是线性的，则会选择最小的正确模型。当未来没有测量误差的协变量可用时，例如由于技术改进或数据收集程序的管理和设计更好，我们的模型选择程序在预测中很有用。
