On sequential estimation

I. A. Ibragimov ; R. Z. Khas'minskii

Teoriâ veroâtnostej i ee primeneniâ, Tome 19 (1974) no. 2, pp. 245-255

Résumé

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.