The Annals of Statistics

Asymptotic Local Minimaxity in Sequential Point Estimation

Michael Woodroofe

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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.

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Ann. Statist., Volume 13, Number 2 (1985), 676-688.

First available in Project Euclid: 12 April 2007

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Primary: 62L12: Sequential estimation

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


Woodroofe, Michael. Asymptotic Local Minimaxity in Sequential Point Estimation. Ann. Statist. 13 (1985), no. 2, 676--688. doi:10.1214/aos/1176349547.

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  • See Correction: Michael Woodroofe. Correction: Asymptotic Local Minimaxity in Sequential Point Estimation. Ann. Statist., Volume 17, Number 1 (1989), 452--452.