A class of admissible estimators of multiple regression coefficient with an unknown variance

Suppose that we observe ${\boldsymbol y} \mid {\boldsymbol \theta },\ \tau \sim N_p({\boldsymbol X}{\boldsymbol \theta },\tau ^{-1}{\boldsymbol I}_p) $y∣θ, τ∼Np(Xθ,τ−1Ip), where ${\boldsymbol \theta } $θ is an unknown vector with unknown precision τ. Estimating the regression coefficient ${\boldsymbol \theta } $θ with known τ has been well studied. However, statistical properties such as admissibility in estimating ${\boldsymbol \theta } $θ with unknown τ are not well studied. Han [(2009). Topics in shrinkage estimation and in causal inference (PhD thesis). Warton School, University of Pennsylvania] appears to be the first to consider the problem, developing sufficient conditions for the admissibility of estimating means of multivariate normal distributions with unknown variance. We generalise the sufficient conditions for admissibility and apply these results to the normal linear regression model. 2-level and 3-level hierarchical models with unknown precision τ are investigated when a standard class of hierarchical priors leads to admissible estimators of ${\boldsymbol \theta } $θ under the normalised squared error loss. One reason to consider this problem is the importance of admissibility in the hierarchical prior selection, and we expect that our study could be helpful in providing some reference for choosing hierarchical priors.