辅助学习及其统计学理解
Auxiliary Learning and its Statistical Understanding
AI总结：
本文针对高维小样本估计难题，提出利用共享协变量的辅助任务构建加权最小二乘估计量，推导最优权重并扩展至广义线性模型，且通过数值实验与智能售货机实例验证。
摘要：
现代统计分析经常遇到高维问题，但样本量却有限。这给传统统计估计方法带来了巨大挑战。在本研究中，我们采用辅助学习来解决高维情形下的估计问题。我们从线性回归设置入手。为了提高主要任务参数估计量的统计效率，我们考虑了若干辅助任务，这些任务与主要任务共享相同的协变量。随后，我们为主要任务构建了一个加权估计量，它是主要任务和辅助任务的普通最小二乘估计量的线性组合。我们解析地推导了最优权重，并研究了相应加权估计量的统计性质。然后，我们将该加权估计量扩展到广义线性回归模型。我们开展了大量数值实验以验证理论结果。最后，为进行说明，我们给出了一个与深度学习相关的智能自动售货机真实数据示例。
英文摘要：
Modern statistical analysis often encounters high-dimensional problems but with a limited sample size. It poses great challenges to traditional statistical estimation methods. In this work, we adopt auxiliary learning to solve the estimation problem in high-dimensional settings. We start with the linear regression setup. To improve the statistical efficiency of the parameter estimator for the primary task, we consider several auxiliary tasks, which share the same covariates with the primary task. Then a weighted estimator for the primary task is developed, which is a linear combination of the ordinary least squares estimators of both the primary task and auxiliary tasks. The optimal weight is analytically derived and the statistical properties of the corresponding weighted estimator are studied. We then extend the weighted estimator to generalized linear regression models. Extensive numerical experiments are conducted to verify our theoretical results. Last, a deep learning-related real-data example of smart vending machines is presented for illustration purposes.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
