On admissible estimation of a mean vector when the scale is unknown

We consider admissibility of generalized Bayes estimators the mean a multivariate normal distribution when scale is unknown under quadratic loss. The priors considered put improper invariant prior on while has hierarchical structure conditional scale. This essentially that Maruyama and Strawderman (2021, Biometrika) (MS21) which indexed by hyperparameter $a$. In paper $a$ chosen so this proper corresponds to $a>-1$. extends MS21 considering with in closed interval $[-2, -1]$, establishing for such authors, (2017, JMVA), have earlier shown $a < -2$ lead inadmissible estimators. therefore completes determination admissibility/inadmissibility class priors. It establishes boundary as = -2$, holding $a\geq inadmissibility -2$. exactly known case these priors, follows from Brown (1971, AOMS). As notable benefit enlargement admissible estimators, we give minimax all dimensions greater than $2$ opposed required dimension be $4$. one particularly interesting special case, establish joint Stein leads estimator $2$.