Open Access
Translator Disclaimer
June, 1985 Asymptotic Local Minimaxity in Sequential Point Estimation
Michael Woodroofe
Ann. Statist. 13(2): 676-688 (June, 1985). DOI: 10.1214/aos/1176349547


Let $X_1, X_2, \cdots$ be i.i.d. random variables with mean $\theta$ and finite, positive variance $\sigma^2,$ depending on unknown parameters $\omega\in\Omega.$ The problem addressed is that of finding a stopping time $t$ for which the risk $R_A(t, \omega) = E_\omega\{A \gamma^2_0(\omega)(\bar{X}_t - \theta)^2 + t\}$ is as small as possible (in a suitable sense), where $A > 0, \gamma_0$ is a positive function on $\Omega$, and $\bar{X}_t = (X_1 + \cdots + X_t)/t.$ For fixed (nonrandom) sample sizes, $2 \sqrt{A}(\gamma_0\sigma)$ is a lower bound for $R_A(n, \omega), n \geq 1$; and the regret of a stopping time $t$ is defined to be $r_A(t, \omega) = R_A(t, \omega) - 2\sqrt{A}(\gamma_0 \sigma).$ The main results determine an asymptotic lower bound, as $A \rightarrow\infty,$ for the minimax regret $M_A(\Omega_0) = \inf_t\sup_{\omega\in\Omega_0}r_A(t, \omega)$ for neighborhoods $\Omega_0$ of arbitrary parameter points $\omega_0 \in \Omega.$ The bound is obtained for multiparameter exponential families and the nonparametric case. The bound is attained asymptotically by an intuitive procedure in several special cases.


Download Citation

Michael Woodroofe. "Asymptotic Local Minimaxity in Sequential Point Estimation." Ann. Statist. 13 (2) 676 - 688, June, 1985.


Published: June, 1985
First available in Project Euclid: 12 April 2007

zbMATH: 0581.62067
MathSciNet: MR790565
Digital Object Identifier: 10.1214/aos/1176349547

Primary: 62L12

Keywords: Bayes risk , exponential families , regret , the Martingale Convergence Theorem , the Minimax Theorem , the nonparametric case , Weighted squared error loss

Rights: Copyright © 1985 Institute of Mathematical Statistics


Vol.13 • No. 2 • June, 1985
Back to Top