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Teor. Veroyatnost. i Primenen., 1971, Volume 16, Issue 2, Pages 339–345 (Mi tvp2158)  

This article is cited in 7 scientific papers (total in 7 papers)

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:

Received: 06.07.1970

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}


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    Citing articles on Google Scholar: Russian citations, English citations
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    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  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    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  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    3. V. N. Solov'ev, “Dual extremal problems and their applications to minimax estimation problems”, Russian Math. Surveys, 52:4 (1997), 685–720  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    4. B. S. Darhovsky, “A new approach to the stochastic recovery problem”, Theory Probab. Appl., 49:1 (2005), 51–64  mathnet  crossref  crossref  mathscinet  zmath  isi
    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  crossref  isi
    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  elib
    7. S. A. Bulgakov, V. M. Khametov, “Vosstanovlenie kvadratichno integriruemoi funktsii po nablyudeniyam s gaussovskimi oshibkami”, UBS, 54 (2015), 45–65  mathnet  elib
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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