An Approach to Inference Following Model Selection With Applications to Transformation-Based and Adaptive Inference
提出一种决策理论框架，用于在模型选择后进行推断（包括检验、点估计和置信估计），并应用于基于变换的推断和自适应推断，解决相关争议。
Abstract A decision-theory approach is formulated for inference following model selection. Inference here refers to either testing, point estimation, or confidence estimation. The loss function includes components for model selection as well as for inference and allows for flexibility in emphasis on one or the other, if such emphasis is desired. A general prescription for Bayes and generalized Bayes procedures is given. A procedure consists of model selection and inference. The general formulation is applied to transformation-based inference, where model selection is equated to choice of transformation. Hinkley and Runger (1984) did transformation-based inference that has aroused controversy. The formulation here is directed to some of these issues. In this approach we explicitly define the quantity of interest for which an inference is desired. Furthermore, we evaluate procedures properly. An example is given where one is interested in estimating the mean of the model selected and the choice of models is either lognormal or gamma. The Hinkley—Runger method and our method are compared. Hogg, Uthoff, Randles, and Davenport (1972) did adaptive inference. They selected from among a finite number of possible location-scale families and then estimated the location of the chosen family. Our formulation is appropriate for this situation. We address the issue of whether Hogg's intuitive adaptive procedure is generalized Bayes and/or admissible for a suitable loss function. We offer a loss function for which a modification of Hogg's procedure has such optimality properties.