策略数据源下线性回归估计的渐近退化
Asymptotic Degradation of Linear Regression Estimates With Strategic Data Sources
摘要：
我们考虑从具有公共利益成分的战略数据源进行线性回归的问题，即，当数据由策略代理提供时，这些策略代理寻求最小化单个提供成本以增加其数据的精度，同时受益于模型的整体精度。与以前的作品相比，我们的模型处理的情况下，有不确定性的属性特征的代理人的数据-一个关键方面的问题时，代理人的数量很大。我们提供了一个表征的游戏的均衡，这揭示了一个有趣的连接与最优设计。随后，我们集中在线性回归参数估计的广义最小二乘的协方差的渐近行为的数据源的数量变得很大。我们提供了这个协方差矩阵的上限和下限，我们表明，当代理商的供应成本是超线性的，该模型的协方差收敛到零，但相对于几乎所有的学习问题与外源数据的速度较慢。另一方面，如果代理人的供应成本是线性的，这个协方差无法收敛。这表明，即使是广义最小二乘估计的基本性质的一致性是妥协时，数据源是战略。
英文摘要：
We consider the problem of linear regression from strategic data sources with a public good component, i.e., when data is provided by strategic agents who seek to minimize an individual provision cost for increasing their data's precision while benefiting from the model's overall precision. In contrast to previous works, our model tackles the case where there is uncertainty on the attributes characterizing the agents' data -- a critical aspect of the problem when the number of agents is large. We provide a characterization of the game's equilibrium, which reveals an interesting connection with optimal design. Subsequently, we focus on the asymptotic behavior of the covariance of the linear regression parameters estimated via generalized least squares as the number of data sources becomes large. We provide upper and lower bounds for this covariance matrix and we show that, when the agents' provision costs are superlinear, the model's covariance converges to zero but at a slower rate relative to virtually all learning problems with exogenous data. On the other hand, if the agents' provision costs are linear, this covariance fails to converge. This shows that even the basic property of consistency of generalized least squares estimators is compromised when the data sources are strategic.
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