The Annals of Mathematical Statistics

Approximation to Bayes Risk in Compound Decision Problems

Allan Oaten

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Abstract

We consider, simultaneously, $N$ statistical decision problems with identical generic structure: state space $\Omega$, action space $A$, sample space $\mathscr{X}$ and nonnegative loss function $L$ defined on $\Omega \times A \times \mathscr{X}$. With $x = (x_1, \cdots, x_N)$ distributed according to $\prod^N_{i=1} \mathbf{P}_{\theta_i} = \mathbf{P}_\theta$, a compound procedure is a vector, $\mathbf{\phi} = (\phi_1, \cdots, \phi_N)$, such that $\phi_i(\mathbf{x}) \in A$ for each $i$ and $\mathbf{x}$. The risk of the procedure $\mathbf{\phi}$ is $\mathbf{R}(\mathbf{\theta}, \mathbf{\phi}) = N^{-1} \sum^N_{r=1} \mathbf{\int L} (\theta_r, \phi_r(\mathbf{x}), x_r)\mathbf{P_\theta} (d\mathbf{x})$ and the modified regret is $\mathbf{D}(\mathbf{\theta, \phi}) = \mathbf{R}(\mathbf{\theta, \phi}) - R(G)$ where $G$ is the empirical distribution of $\theta_1, \cdots, \theta_N$, and $R(G)$ is the Bayes risk against $G$ in the component problem. We discuss quite wide classes of procedures, $\mathbf{\phi}$, which consist of using $\mathbf{x}$ to estimate $G$, and then playing $\varepsilon$-Bayes against the estimate in each component problem. For one class we establish a type of uniform convergence of the conditional risk in the $m \times n$ problem (i.e. $\Omega$ has $m$ elements, $A$ has $n$), and use this to get $\mathbf{D}(\mathbf{\theta, \phi}) < \varepsilon + o(1)$ for another class in the $m \times n$ and $m \times \infty$ problems. Similar, but weaker, results are given in part II for the case when $\Omega$ is infinite.

Article information

Source
Ann. Math. Statist., Volume 43, Number 4 (1972), 1164-1184.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177692469

Digital Object Identifier
doi:10.1214/aoms/1177692469

Mathematical Reviews number (MathSciNet)
MR312612

Zentralblatt MATH identifier
0241.62005

JSTOR
links.jstor.org

Citation

Oaten, Allan. Approximation to Bayes Risk in Compound Decision Problems. Ann. Math. Statist. 43 (1972), no. 4, 1164--1184. doi:10.1214/aoms/1177692469. https://projecteuclid.org/euclid.aoms/1177692469


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