Approximate Marginal Densities of Coordinate Functions When the Maxima Occurs at the Boundary
Feb 27 1992
Laplacian type expansions are proposed for the marginal densities of coordinate functions θj of an m-dimensional random vector θ ϵ Θ when its joint density has a dominating local mode that lies outside or on the (m-1)-dimensional smooth boundary T of Θ, where Θ is a bounded subset in Rm. Specifically, two kinds of approximations are considered. The first one is similar to Tierney and Kadane (1986) approximation and it employs a conditional maximization over the nuisance variables. It is shown that under regularity conditions this approximation has a relative error of order n-1, which holds uniformly on bounded regions of the marginal variable. the second type of approximation is developed for the cases when the conditional maximization fails, e.g., the location of the maxima lies outside the admissible region of integration. This latter approximation is related to integrating Edgeworth-like series expansions over the nuisance variables and incurs a relative error of order n-1/2, also holds uniformly on bounded regions about the marginal variable. Both methods are illustrated with Bayesian applications.
Keywords:Asymptotics, Bayesian inference, edgeworth expansions, divergeance theorem, laplace's method, saddle-point approximation, Tierney-Kadane approximation