I don't know why the order of error term is $O(n^{-1})$ or more high in asymptotically computing integral using Laplace's method? For instance, the following examples:
Suppose that $h(\theta)$ is a real function, has a unique manimum at $\hat{\theta}$ and has
continuously second derivative, then we get \int_{-\infty}^{+\infty}e^{-nh(\theta)}d\theta=e^{-nh(\hat{\theta})}(\frac{2\pi}{nh^{''}(\hat{\theta})})^{\frac{1}{2}}(1+O(n^{-1})), where h^{''}(\hat{\theta})\neq 0.
But, I have computed this asymptotic expression many times, I still can not get the order of error term is $O(n^{-1}).$ On contrary, I can only get the order is $O(n).$ I do not know why.
There is another question on this expression, which is what is the $n$ that is arbitrary nature number. A lot of literature only use this result give some illustrations and do not give some proofs.
Hence, I consult everyone and wish to get your answers about this questions. Thanks a lot!