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Teoriya Veroyatnostei i ee Primeneniya, 1974, Volume 19, Issue 2, Pages 245–255
(Mi tvp2826)
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This article is cited in 9 scientific papers (total in 9 papers)
On sequential estimation
I. A. Ibragimova, R. Z. Khas'minskiib a Leningrad
b Moscow
Abstract:
The main result of this paper is
Theorem 1. {\em Let $X_1,\dots,X_n$ be independent observations with probability density $f(x,\theta)$, $\theta\in\Theta\subset R^1$. Let the following conditions be satisfied:
1) $f(x,\theta)$ is absolutely continuous as a function of $\theta$ in some neighbourhood of $\theta=t$ for all $x$;
2) for each $\theta$, derivative $\partial f(x,\theta)/\partial\theta$ exists in some neighbourhood of $t$ for $\nu$-almost all $x$;
3) the function $I(\theta)$ (see (1.1)) is continuous at $\theta=t$.
Let $(\{T_m^{(n)}\},\tau_n)$ be a sequential estimation procedure and $E_\theta\tau_n=n$. Then, for any $a>0$, inequality (1.3) holds true.}
This theorem shows that for the loss function $|x|^a$ sequential estimation does not give advantage in the asymptotically minimax sense.
Received: 04.05.1973
Citation:
I. A. Ibragimov, R. Z. Khas'minskii, “On sequential estimation”, Teor. Veroyatnost. i Primenen., 19:2 (1974), 245–255; Theory Probab. Appl., 19:2 (1975), 233–244
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https://www.mathnet.ru/eng/tvp2826 https://www.mathnet.ru/eng/tvp/v19/i2/p245
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