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