Asymptotic expansions associated with some statistical estimates in the smooth case. II. Expansions of moments and distributions

Let $x_1,\dots,x_n$ be a sample from a distribution $\mathbf P_{\theta}$ with density $f(x,\theta)$, $\theta\in \Theta\subset R^1$. Let $T_n$ be a Bayesian estimate or a maximum posterior density estimate. The expansions
$$ \sqrt n(T_n-\theta)=\xi_0+\xi_1\frac{1}{\sqrt n}+\dots+\xi_{k-1}\biggl(\frac{1}{\sqrt n}\biggr)^{k-1}+\widetilde\xi_{k,n}\biggl(\frac{1}{\sqrt n}\biggr)^k, $$
obtained in [1], imply expansions of the moments $\mathbf E_{\theta}(\sqrt n(T_n-\theta))^m$ where $m\ge 1$ is an integer, and expansions of the distribution functions $\mathbf P_{\theta}\{\sqrt n(T_n-\theta)<z\}$. Linnik's problem of calculating the terms of order $1/n$ in the expansion of $\mathbf E_{\theta}(\sqrt n(T_n-\theta))^2$ is solved.