Asymptotic Expansions of Posterior Distributions in a Non-Regular Model

Abstract

Let $f$ be a density function with respect to Lebesgue measure. We suppose that $f(x)>0$ on $(0,\beta)$, where $0<\beta\leqq+\infty$, and $f$ is uniformly continuous on $(0,\beta)$. Moreover, let $f'(x)\to\alpha$ as $x\to +0$ exist, where $0<\alpha<+\infty$. We consider a non-regular model defined by $f(x,\theta)=f(x-\theta)$, $\theta,x\in\mathbf{R}$. In the present paper, under some conditions, it is shown that when $\theta$ is regarded as a random variable with a prior density function with respect to Lebesgue measure, there exist asymptotic expansions of centered and scaled posterior distributions of $\theta$.