摘要
We reexamine the classical linear regression model when it is subject to two types of uncertainty:(i)some covariates are either missing or completely inaccessible,and(ii)the variance of the measurement error is undetermined and changing according to a mechanism unknown to the statistician.By following the recent theory of sublinear expectation,we propose to characterize such mean and variance uncertainty in the response variable by two specific nonlinear random variables,which encompass an infinite family of probability distributions for the response variable in the sense of(linear)classical probability theory.The approach enables a family of estimators under various loss functions for the regression parameter and the parameters related to model uncertainty.The consistency of the estimators is established under mild conditions in the data generation process.Three applications are introduced to assess the quality of the approach including a forecasting model for the S&P Index.
基金
supported by the National Key R&D program of China(Grant Nos.2018YFA0703900 and ZR2019ZD41)
the National Natural Science Foundation of China(Grant No.11701330)
Taishan Scholar Talent Project Youth Project.
