Abstract
For a vast array of general spherically symmetric location-scale models with a residual vector, we consider estimating the (univariate) location parameter when it is lower bounded. We provide conditions for estimators to dominate the benchmark minimax MRE estimator, and thus be minimax under scale invariant loss. These minimax estimators include the generalized Bayes estimator with respect to the truncation of the common non-informative prior onto the restricted parameter space for normal models under general convex symmetric loss, as well as non-normal models under scale invariant \(L^p\) loss with \(p>0\). We cover many other situations when the loss is asymmetric, and where other generalized Bayes estimators, obtained with different powers of the scale parameter in the prior measure, are proven to be minimax. We rely on various novel representations, sharp sign change analyses, as well as capitalize on Kubokawa’s integral expression for risk difference technique. Several properties such as robustness of the generalized Bayes estimators under various loss functions are obtained.
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Notes
It is interesting to point out that the arguments here apply as well to strictly bowled-shaped losses. As well, only a stochastic increasing property for the densities \(f\) is required. The monotonicity of \(g_{\pi _0}\), however, is guaranteed by the convexity of \(\rho \) (Lemma ), which is assumed here.
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Acknowledgments
This work was partially supported by a grant from the Simons Foundation (#209035) to William Strawderman. Mohammad Jafari Jozani and Éric Marchand gratefully acknowledge the research support of the Natural Sciences and Engineering Research Council of Canada. The authors would like to thank referees for careful reading of the paper and useful suggestions.
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Jafari Jozani, M., Marchand, É. & Strawderman, W.E. Estimation of a non-negative location parameter with unknown scale. Ann Inst Stat Math 66, 811–832 (2014). https://doi.org/10.1007/s10463-013-0425-x
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DOI: https://doi.org/10.1007/s10463-013-0425-x