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 Teor. Veroyatnost. i Primenen.: Year: Volume: Issue: Page: Find

 Teor. Veroyatnost. i Primenen., 1976, Volume 21, Issue 1, Pages 16–33 (Mi tvp3272)

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

S. I. Gusev

Abstract: Let $x_1,…,x_n$ 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
$$\sqrt n(T_n-\theta)=\xi_0+\xi_1\frac{1}{\sqrt n}+…+\xi_{k-1}(\frac{1}{\sqrt n})^{k-1}+\widetilde\xi_{k,n}(\frac{1}{\sqrt n})^k,$$
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.

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English version:
Theory of Probability and its Applications, 1976, 21:1, 14–33

Bibliographic databases:

Citation: S. I. Gusev, “Asymptotic expansions associated with some statistical estimates in the smooth case. II. Expansions of moments and distributions”, Teor. Veroyatnost. i Primenen., 21:1 (1976), 16–33; Theory Probab. Appl., 21:1 (1976), 14–33

Citation in format AMSBIB
\Bibitem{Gus76} \by S.~I.~Gusev \paper Asymptotic expansions associated with some statistical estimates in the smooth case. II. Expansions of moments and distributions \jour Teor. Veroyatnost. i Primenen. \yr 1976 \vol 21 \issue 1 \pages 16--33 \mathnet{http://mi.mathnet.ru/tvp3272} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=413327} \zmath{https://zbmath.org/?q=an:0403.62020} \transl \jour Theory Probab. Appl. \yr 1976 \vol 21 \issue 1 \pages 14--33 \crossref{https://doi.org/10.1137/1121002} 

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This publication is cited in the following articles:
1. M. V. Burnashev, “Investigation of second order properties of statistical estimators in a scheme of independent observations”, Math. USSR-Izv., 18:3 (1982), 439–467
2. A. A. Zaikin, “Asymptotic expansion of posterior distribution of parameter centered by a $\sqrt n$-consistent estimate”, J. Math. Sci. (N. Y.), 229:6 (2018), 678–697
3. Zaikin A.A., “Asymptotic Expansion of D-Risks For Hypothesis Testing in Bernoulli Scheme”, Lobachevskii J. Math., 39:3, SI (2018), 413–423