Abstract
Problem statement: Wald showed that the minimax solution is the Bayes ian solution with respect to the law a priori the worst. We try to establish a similar result by comparing the Bayesian solution and the solution of maximum likel ihood when the parameter space is a compact metrizable group. Approach: we take as a priori law Haar measure because we re duce the problem by invariance. We construct a sequence of cost func tions for which we obtain a sequence of solutions Bayesian which converges to the solution of the maximum likelihood. Results: We show that both solutions are asymptotically equal. Conclusion/Recommendation : The generalization when the parameter space is a local compact group.
Highlights
Problem position: The fundamental problem of statistical decision theory can be summarized as follows: Given the triplet (Θ, D, C)
An operation of G on E is a mapping: Topological groups: Just like that pointed out in the introduction, command us led to study a priori law meeting ownership of invariance; the importance to remind of some notions on the topological groups and of the measure of Haar
Haar measure notations: Let G be a topological group operating continuously to left on a local level compact
Summary
Problem position: The fundamental problem of statistical decision theory can be summarized as follows: Given the triplet (Θ, D, C). Definition 2.1.5: Let two measurable spaces (Ω, α) and (Θ, τ) and (Pθ), θ∈Θ a family of likelihood of transition on (Ω, α) defined on (Θ, τ). An operation (either operation to the left or action) of G on E is a mapping: Topological groups: Just like that pointed out in the introduction, command us led to study a priori law meeting ownership of invariance; the importance to remind of some notions on the topological groups and of the measure of Haar It is the aim of the present section. Haar measure notations: Let G be a topological group operating continuously to left on a local level compact. Definition 2.4.4: Let G be a locally compact group Haar measure we call the left (respectively right) of G a positive non-zero, σ- fine G invariant on the left We denote by L, (θ) the probability density pθ par rapport to Q
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