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

TL;DR

This paper proves the admissibility of the standard confidence set for one or two regression coefficients in the Gaussian linear regression model with unknown variance, resolving a long-standing open problem and impacting modern inference methods.

Contribution

It establishes the admissibility of the usual confidence set in the unknown-variance case for small numbers of parameters, and introduces a new class of conjugate priors for Gaussian models.

Findings

Standard confidence set is admissible for 1 or 2 regression coefficients.

Solves a long-standing open problem in statistical inference.

Introduces a new class of conjugate priors for Gaussian models.

Abstract

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.