The Annals of Statistics

Asymptotic Local Minimaxity in Sequential Point Estimation

Michael Woodroofe

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Abstract

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.

Article information

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

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176349547

Digital Object Identifier
doi:10.1214/aos/1176349547

Mathematical Reviews number (MathSciNet)
MR790565

Zentralblatt MATH identifier
0581.62067

JSTOR
links.jstor.org

Subjects
Primary: 62L12: Sequential estimation

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

Citation

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


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Corrections

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