Laplace approximations using nα-consistent estimators

When applying Laplace's method to approximate an integral, we require an exact mode of its integrand. In this paper, we derive a Laplace approximation using an asymptotic mode of order Op(n−α)(α≥1∕2) instead of the exact mode, and present some ways to establish asymptotic modes with a suitable order. The class of the asymptotic modes is wide and includes nα-consistent estimators (α≥1∕2) and a true parameter. Second-order approximations and asymptotic properties to posterior expectations of smooth functions are derived by the proposed approximation.