|
Teoriya Veroyatnostei i ee Primeneniya, 1971, Volume 16, Issue 2, Pages 339–345
(Mi tvp2158)
|
|
|
|
This article is cited in 30 scientific papers (total in 30 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 $f(t)=a_0+a_1t+\dots+ a_nt^n+\mathrm g(t)t^{n+1}$ 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.
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
Linking options:
https://www.mathnet.ru/eng/tvp2158 https://www.mathnet.ru/eng/tvp/v16/i2/p339
|
|