We discuss minimum mean squared error and Bayesian estimation of the variance and its
common transformations in the setting of normality and homoscedasticity with small samples, for
which asymptotics do not apply. We show that permitting some bias can be rewarded by greatly
reduced mean squared error. We apply borderline and equilibrium priors. The purpose of these
priors is to reduce the onus on the expert or client to specify a single prior distribution that would
capture the information available prior to data inspection. Instead, the (parametric) class of all
priors considered is partitioned to subsets that result in the preference for different actions. With
the family of conjugate inverse gamma priors, this Bayesian approach can be formulated in the
frequentist paradigm, describing the prior as being equivalent to additional observations.Peer Reviewe
