Stable Decision Problems

Stable Decision Problems

A decision problem is characterized by a loss function V and opinion H. The pair (V, H) is said to be strongly stable iff for every sequence Fn→ωH, Gn→ωH and Ln→ V, Wn→ V uniformly, $\lim_{\varepsilon \downarrow 0} \lim \sup_{n\rightarrow \infty} \lbrack \int L_n(\theta, D_n(\varepsilon)) dF_n(\thet...

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Bibliographische Detailangaben
Veröffentlicht in:The Annals of Statistics. - Institute of Mathematical Statistics. - 6(1978), 5, Seite 1095-1110
1. Verfasser: Kadane, Joseph B. (VerfasserIn)
Weitere Verfasser: Chuang, David T.
Format: Online-Aufsatz
Sprache:English
Veröffentlicht: 1978
Zugriff auf das übergeordnete Werk:The Annals of Statistics
Schlagworte:Decision theory robustness stable estimation stable decisions Mathematics Philosophy Behavioral sciences Business
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520 |a A decision problem is characterized by a loss function V and opinion H. The pair (V, H) is said to be strongly stable iff for every sequence Fn→ωH, Gn→ωH and Ln→ V, Wn→ V uniformly, $\lim_{\varepsilon \downarrow 0} \lim \sup_{n\rightarrow \infty} \lbrack \int L_n(\theta, D_n(\varepsilon)) dF_n(\theta) - \inf_D \int L_n(\theta, D) dF_n(\theta)\rbrack = 0$ for every sequence Dn(ε) satisfying $\int W_n(\theta, D_n(\varepsilon)) dG_n(\theta) \leqq \inf_D \int W_n(\theta, D) dG_n(\theta) + \varepsilon.$ We show that squared error loss is unstable with any opinion if the parameter space is the real line and that any bounded loss function V(θ, D) that is continuous in θ uniformly in D is stable with any opinion H. Finally we examine the estimation or prediction case V(θ, D) = h(θ - D), where h is continuous, nondecreasing in (0, ∞) and nonincreasing in (-∞, 0) and has bounded growth. While these conditions are not enough to assure strong stability, various conditions are given that are sufficient. We believe that stability offers the beginning of a Bayesian theory of robustness. 
540 |a Copyright 1978 Institute of Mathematical Statistics 
650 4 |a Decision theory 
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650 4 |a Philosophy  |x Logic  |x Logical argument  |x Counterexamples 
650 4 |a Behavioral sciences  |x Psychology  |x Cognitive psychology  |x Cognitive processes  |x Decision making  |x Bayesian theories 
650 4 |a Behavioral sciences  |x Psychology  |x Cognitive psychology  |x Decision theory 
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