Admissible Selection of an Accurate and Parsimonious Normal Linear Regression Model

Let M₀ be a normal linear regression model and let M₁,⋯, MK be distinct proper linear submodels of M₀. Let k ∈ \0,⋯, K\ be a model selection rule based on observed data from the true model. Given k, let the unknown parameters of the selected model Mₖ be fitted by the maximum likelihood method. A loss function is introduced which depends additively on two parts: (i) a measure of the difference between the fitted model Mₖ and the true model; and (ii) a measure Cₖ of the "complexity" of the selected model. A natural model selection rule k̄, which minimizes an empirical version of this loss, is shown to be admissible and very nearly Bayes.