Semiparametric posterior limits under local asymptotic exponentiality

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 conditions on model and prior comparable to those of Bickel and Kleijn (2012), the posterior distribution displays Bernstein-von Mises-type asymptotic behaviour, with exponential distributions as the limiting sequence. Results are applied to semiparametric LAE location and scaling examples, with the potential for semiparametric bias and a discussion of Bayesian methods to avoid it.