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Asymptotic admissibility of priors and elliptic differential equations. (English) Zbl 1326.62020

Fourdrinier, Dominique (ed.) et al., Contemporary developments in Bayesian analysis and statistical decision theory. A festschrift for William E. Strawderman. Beachwood, OH: IMS, Institute of Mathematical Statistics (ISBN 978-0-940600-81-2). Institute of Mathematical Statistics Collections 8, 117-130 (2012).
Summary: We evaluate priors by the second order asymptotic behaviour of the corresponding estimators. Under certain regularity conditions, the risk differences between efficient estimators of parameters taking values in a domain \(D\), an open connected subset of \(\mathbb{R}^d\), are asymptotically expressed as elliptic differential forms depending on the asymptotic covariance matrix \(V\). Each efficient estimator has the same asymptotic risk as a “local Bayes” estimate corresponding to a prior density \(p\). The asymptotic decision theory of the estimators identifies the smooth prior densities as admissible or inadmissible, according to the existence of solutions to certain elliptic differential equations. The prior \(p\) is admissible if the quantity \(pV\) is sufficiently small near the boundary of \(D\). We exhibit the unique admissible invariant prior for \(V=I\), \(D=\mathbb{R}^d-\{0\}\). A detailed example is given for a normal mixture model.
For the entire collection see [Zbl 1319.62003].

MSC:

62C15 Admissibility in statistical decision theory
62F10 Point estimation
62F12 Asymptotic properties of parametric estimators
62F15 Bayesian inference
62J07 Ridge regression; shrinkage estimators (Lasso)
62P30 Applications of statistics in engineering and industry; control charts
35J15 Second-order elliptic equations
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