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Change-point estimation: Linear statistics and asymptotic Bayes risk. (English) Zbl 0884.62028

Summary: In the retrospective setting of the change-point estimation problem a class of linear estimators, based on a scalar statistic, is introduced, and the Bayes estimator within this class is obtained. The asymptotic properties of this estimator are obtained via the theory of integral operators in Hilbert space. The asymptotic expansion of the minimum Bayes risk is found and its leading term is related to the mixture parameter estimation problem. On the basis of this formula the optimal choice of the underlying statistic is suggested. The asymptotic admissibility and the minimaxity of the unbiased linear estimator are established. Explicit formulas are obtained for the uniform prior distribution when variances before and after change are equal.

MSC:

62F12 Asymptotic properties of parametric estimators
62C10 Bayesian problems; characterization of Bayes procedures
62E20 Asymptotic distribution theory in statistics
62F15 Bayesian inference
47G10 Integral operators
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