Admissibility of the usual confidence set for the mean of a univariate or bivariate normal population: The unknown-variance case

In the Gaussian linear regression model (with unknown mean and variance), we show that the standard confidence set for one or two regression coefficients is admissible in the sense of Joshi (1969). This solves a long-standing open problem in mathematical statistics, and this has important implications on the performance of modern inference procedures post-model-selection or post-shrinkage, particularly in situations where the number of parameters is larger than the sample size. As a technical contribution of independent interest, we introduce a new class of conjugate priors for the Gaussian location-scale model.