Teoriya Veroyatnostei i ee Primeneniya
 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., 1971, Volume 16, Issue 2, Pages 339–345 (Mi tvp2158)

Short Communications

Minimax weights in a trend detection problem for a stochastic process

I. L. Legostaeva, A. N. Širyaev

Moscow

Abstract: Let $F_n(M)$ be the class of real functions of the form $f(t)=a_0+a_1t+…+ a_nt^n+\mathrm g(t)t^{n+1}$ where $\sup\limits_t|\mathrm g(t)|\le M$, $-\infty<t<\infty$.
The problem considered is to estimate the regression coefficient $a_0=f(0)$ from the data $\xi(t)=f(t)+\eta(t)$, $\eta(t)$ being a white noise process ($\mathbf M\eta(t)=0$, $\mathbf M\eta(s)\eta(t)=d^2\delta(t-s)$). For the class of linear estimators $\widehat f(0)=\int_{-\infty}^\infty l(t)\xi(t) dt$, a weight $l^*(t)$ is called minimax if
$$\sup_{f\in F_n(M)}\Delta(l^*,f)=\inf_l\sup_{f\in F_n(M)}\Delta(l,f)$$
where $\Delta(l,f)=\mathbf M[f(0)-\widehat f(0)]^2$.
Theorem 1 gives necessary and sufficient conditions for a weight to be minimax. For $n=0$ and $n=1$ minimax weights are obtained in Theorem 2.

Full text: PDF file (367 kB)

English version:
Theory of Probability and its Applications, 1971, 16:2, 344–349

Bibliographic databases:

Citation: I. L. Legostaeva, A. N. Širyaev, “Minimax weights in a trend detection problem for a stochastic process”, Teor. Veroyatnost. i Primenen., 16:2 (1971), 339–345; Theory Probab. Appl., 16:2 (1971), 344–349

Citation in format AMSBIB
\Bibitem{LegShi71} \by I.~L.~Legostaeva, A.~N.~{\v S}iryaev \paper Minimax weights in a~trend detection problem for a~stochastic process \jour Teor. Veroyatnost. i Primenen. \yr 1971 \vol 16 \issue 2 \pages 339--345 \mathnet{http://mi.mathnet.ru/tvp2158} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=292220} \zmath{https://zbmath.org/?q=an:0237.62060} \transl \jour Theory Probab. Appl. \yr 1971 \vol 16 \issue 2 \pages 344--349 \crossref{https://doi.org/10.1137/1116031} 

• http://mi.mathnet.ru/eng/tvp2158
• http://mi.mathnet.ru/eng/tvp/v16/i2/p339

 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. R. D. Dodunekova, “Minimax estimation in problems with incomplete data”, Russian Math. Surveys, 39:1 (1984), 145–146
2. V. N. Solov'ev, “A minimax-Bayes estimate on classes of distributions with bounded second moments”, Russian Math. Surveys, 50:4 (1995), 832–834
3. V. N. Solov'ev, “Dual extremal problems and their applications to minimax estimation problems”, Russian Math. Surveys, 52:4 (1997), 685–720
4. B. S. Darhovsky, “A new approach to the stochastic recovery problem”, Theory Probab. Appl., 49:1 (2005), 51–64
5. Darkhovskiy B., “Non Asymptotic Minimax Estimation of Functionals with Noisy Observations”, Communications in Statistics-Simulation and Computation, 41:6, Part 1 Sp. Iss. SI (2012), 787–803
6. Korovin Ya.S., Tkachenko M.G., Kononov S.V., “Operativnaya diagnostika sostoyaniya neftepromyslovogo oborudovaniya na osnove tekhnologii intellektualnoi obrabotki dannykh”, Neftyanoe khozyaistvo, 2012, no. 9, 116–118
7. S. A. Bulgakov, V. M. Khametov, “Vosstanovlenie kvadratichno integriruemoi funktsii po nablyudeniyam s gaussovskimi oshibkami”, UBS, 54 (2015), 45–65