2021/08/04 by Shuzhen Yang, Jianfeng Yao, Yang, Shuzhen +1 · 2 citations
Mathematics · #Advanced Statistical Methods and Models #FOS: Mathematics #Fuzzy Systems and Optimization #Statistical Methods and Inference #Statistics Theory (math.ST) #math.ST #stat.TH
paper · pdf · doi:10.48550/arxiv.2108.02140
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arxiv created 2021/08/04 · openalex publication_date 2021/08/04 · arxiv updated 2021/08/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We reexamine the classical linear regression model when the model is subject to two types of uncertainty: (i) some of 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 on the data generation process. Three applications are introduced to assess the quality of the approach including a forecasting model for the S&P Index.