RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Subscription Guidelines for authors Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

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

 Teor. Veroyatnost. i Primenen., 1973, Volume 18, Issue 2, Pages 303–311 (Mi tvp4248)

An Asymptotic Expansion for a Class of Estimators Containing Maximum Likelihood Estimators

D. M. Chibisov

Moscow

Abstract: Let $X_1,…,X_n$ be a sample from a distribution dependent on a parameter $\theta=(\theta_1,…,\theta_s)^T$ and $\vartheta_n=\vartheta_n(X_1,…,X_n)$ a minimum contrast estimator for $\theta$ corresponding to a contrast function $f(x,\theta)$ (see, e.g., [7], [8], [9]). When the $X_i$'s have a density $p(x,\theta)$ and $f(x,\theta)=-\log p(x,\theta), \vartheta$ is the maximum likelihood estimator. Among the regularity conditions, it is assumed that the continous derivatives $f^{\alpha}(x,\theta)=(d^{\alpha_1+…+\alpha_s}/d\theta_1^{\alpha_1}…d\theta_s^{\alpha_s})f(x,\theta)$ exist in a neighbourhood of the true value $\theta_0$ for all $\alpha$ with $\alpha_1+…+\alpha_s\le k+1$ and $E_{\theta_0}|f^{(\alpha)}(X,\theta_0)|^r<\infty$ for some $r>2$. We obtain an expansion of the form
$$\sqrt{n}(\vartheta_n-\theta_0)=h_1+n^{-1/2}h_2+…+n^{-(k-1)/2}h_k+\zeta_n$$
where the components of $h_j, j=1,\ldots,k$, are polynomials dependent on some random variables of the form $n^{-1/2}\sum\limits_{i=1}^n f^{(\alpha)}(X_i,\theta_0)$ and $\zeta_n$ is a random variable converging to zero at a certain rate.

Full text: PDF file (1216 kB)

English version:
Theory of Probability and its Applications, 1973, 18:2, 295–303

Bibliographic databases:

Citation: D. M. Chibisov, “An Asymptotic Expansion for a Class of Estimators Containing Maximum Likelihood Estimators”, Teor. Veroyatnost. i Primenen., 18:2 (1973), 303–311; Theory Probab. Appl., 18:2 (1973), 295–303

Citation in format AMSBIB
\Bibitem{Chi73} \by D.~M.~Chibisov \paper An Asymptotic Expansion for a Class of Estimators Containing Maximum Likelihood Estimators \jour Teor. Veroyatnost. i Primenen. \yr 1973 \vol 18 \issue 2 \pages 303--311 \mathnet{http://mi.mathnet.ru/tvp4248} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=321227} \zmath{https://zbmath.org/?q=an:0295.62028} \transl \jour Theory Probab. Appl. \yr 1973 \vol 18 \issue 2 \pages 295--303 \crossref{https://doi.org/10.1137/1118031} 

• http://mi.mathnet.ru/eng/tvp4248
• http://mi.mathnet.ru/eng/tvp/v18/i2/p303

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. M. V. Burnashev, “Asymptotic expansions for estimates of a signal parameter in Gaussian white noise”, Math. USSR-Sb., 33:2 (1977), 159–184
2. 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
3. E. E. Zhuk, “Cluster Analysis of the Realizations of Autoregression Time Series”, Autom. Remote Control, 64:1 (2003), 65–75
•  Number of views: This page: 182 Full text: 80