Collaborative Research: Prior-free probabilistic inferential methods for "large-p-small-n" linear regression problems

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The investigators study prior-free probabilistic inference with "large p, small n" regression analysis. This is made possible in the new framework of Inferential Models (IMs) proposed recently by the investigators. Statistical results produced by IMs are probabilistic and have desirable frequency properties. In this study, the investigators develop IM-based methods for linear and certain non-linear regression analysis. A sequence of topics in the context of large-p-small-n regression to be investigated include (1) variable selection in Gaussian regression models; (2) robust Student-t regression; and (3) binary regression models.Linear regression is one of the most commonly used methodologies in statistical applications. However, desirable prior-free and frequency-calibrated probabilistic inference, particularly in the important variable selection context, has not been available until the recent development of IMs. The IM framework provides a new and promising alternative to the well-known Bayesian and frequentist methods for various high-dimensional problems researchers currently face. In this study, the investigators develop new statistical methods and computing software, generating useful tools for applied statisticians and scientists who are challenged by very-high-dimensional data in carrying out regression analysis.