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On the equivariance criterion in statistical prediction

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  • Haojin Zhou
  • Tapan Nayak

Abstract

This paper presents a general development of the basic logic of equivariance for a parametric point prediction problem. We propose a framework that allows the set of possible predictions as well as the losses to depend on the data and then explore the nature and properties of relevant transformation groups for applying the functional and formal equivariance principles. We define loss invariance and predictive equivariance appropriately and discuss their ramifications. We describe a structure of equivariant predictors in terms of maximal invariants and present a method for deriving minimum risk equivariant predictors. We explore the connections between equivariance and risk unbiasedness and show that uniquely best risk unbiased predictors are almost equivariant. We apply our theoretical results to some illustrative examples. Copyright The Institute of Statistical Mathematics, Tokyo 2015

Suggested Citation

  • Haojin Zhou & Tapan Nayak, 2015. "On the equivariance criterion in statistical prediction," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 67(3), pages 541-555, June.
    • Handle: RePEc:spr:aistmt:v:67:y:2015:i:3:p:541-555
    DOI: 10.1007/s10463-014-0464-y
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    References listed on IDEAS

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    1. J. F. Lawless & Marc Fredette, 2005. "Frequentist prediction intervals and predictive distributions," Biometrika, Biometrika Trust, vol. 92(3), pages 529-542, September.
    2. Yushan Xiao, 2000. "Linex Unbiasedness in a Prediction Problem," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 52(4), pages 712-721, December.
    3. Deshpande, Jayant V. & Muhammad Fareed, T. P., 1995. "A note on conditionally unbiased estimation after selection," Statistics & Probability Letters, Elsevier, vol. 22(1), pages 17-23, January.
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