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March, 1990 On Minimax Estimation in the Presence of Side Information About Remote Data
R. Ahlswede, M. V. Burnashev
Ann. Statist. 18(1): 141-171 (March, 1990). DOI: 10.1214/aos/1176347496

Abstract

We analyze the following model: One person, called "helper" observes an outcome $x^n = (x_1, \cdots, x_n) \in \mathscr{X}^n$ of the sequence $X^n = (X_1, \cdots, X_n)$ of i.i.d. RV's and the statistician gets a sample $y^n = (y_1, \cdots, y_n)$ of the sequence $Y^n(\theta, x^n)$ of RV's with a density $\prod^n_{t = 1} f(y_t \mid \theta, x_t)$. The helper can give some (side) information about $x^n$ to the statistician via an encoding function $s_n: \mathscr{X}^n \rightarrow \mathbb{N}$ with rate($s_n)^{def}{=}(1/n)\log {\tt\#}$ range($s_n) \leq R$. Based on the knowledge of $s_n(x^n)$ and $y^n$ the statistician tries to estimate $\theta$ by an estimator $\hat{\theta}_n$. For the maximal mean square error $e_n(R) =^{def} \inf_{\hat\theta_n} \inf_{s_n: \text{rate}}(s_n) \leq R \sup_{\theta \in \Theta} E_\theta|\hat{\theta}_n - \theta|^2$ we establish a Cramer-Rao type bound and, in case of a finite $\mathscr{X}$, prove asymptotic achievability of this bound under certain conditions. The proof involves a nonobvious combination of results (some of which are novel) for both coding and estimation.

Citation

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R. Ahlswede. M. V. Burnashev. "On Minimax Estimation in the Presence of Side Information About Remote Data." Ann. Statist. 18 (1) 141 - 171, March, 1990. https://doi.org/10.1214/aos/1176347496

Information

Published: March, 1990
First available in Project Euclid: 12 April 2007

zbMATH: 0712.62023
MathSciNet: MR1041389
Digital Object Identifier: 10.1214/aos/1176347496

Subjects:
Primary: 62A99
Secondary: 62F12, 62N99, 94A15

Rights: Copyright © 1990 Institute of Mathematical Statistics

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Vol.18 • No. 1 • March, 1990
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