  Consider semiparametric models that display local asymptotic exponentiality
(Ibragimov and Has'minskii (1981)), an asymptotic property of the likelihood
associated with discontinuities of densities. Our interest goes to estimation
of the location of such discontinuities while other aspects of the density form
a nuisance parameter. It is shown that under certain conditions on model and
prior, the posterior distribution displays Bernstein-von Mises-type asymptotic
behaviour, with exponential distributions as the limiting sequence. In contrast
to regular settings, the maximum likelihood estimator is inefficient under this
form of irregularity. However, Bayesian point estimators based on the limiting
posterior distribution attain the minimax risk. Therefore, the limiting
behaviour of the posterior is used to advocate efficiency of Bayesian point
estimation rather than compare it to frequentist estimation procedures based on
the maximum likelihood estimator. Results are applied to semiparametric LAE
location and scaling examples.
