On geometry of the set of admissible quadratic estimators of quadratic functions of normal parameters

Konrad Neumann; Stefan Zontek

Discussiones Mathematicae Probability and Statistics (2006)

  • Volume: 26, Issue: 2, page 109-125
  • ISSN: 1509-9423

Abstract

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We consider the problem of admissible quadratic estimation of a linear function of μ² and σ² in n dimensional normal model N(Kμ,σ²Iₙ) under quadratic risk function. After reducing this problem to admissible estimation of a linear function of two quadratic forms, the set of admissible estimators are characterized by giving formulae on the boundary of the set D ⊂ R² of components of the two quadratic forms constituting the set of admissible estimators. Different shapes and topological properties of the set D are studied.

How to cite

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Konrad Neumann, and Stefan Zontek. "On geometry of the set of admissible quadratic estimators of quadratic functions of normal parameters." Discussiones Mathematicae Probability and Statistics 26.2 (2006): 109-125. <http://eudml.org/doc/277042>.

@article{KonradNeumann2006,
abstract = {We consider the problem of admissible quadratic estimation of a linear function of μ² and σ² in n dimensional normal model N(Kμ,σ²Iₙ) under quadratic risk function. After reducing this problem to admissible estimation of a linear function of two quadratic forms, the set of admissible estimators are characterized by giving formulae on the boundary of the set D ⊂ R² of components of the two quadratic forms constituting the set of admissible estimators. Different shapes and topological properties of the set D are studied.},
author = {Konrad Neumann, Stefan Zontek},
journal = {Discussiones Mathematicae Probability and Statistics},
keywords = {linear estimator; quadratic estimator; Bayesian quadratic estimator; quadratic loss function; admissibility; quadratic subspace},
language = {eng},
number = {2},
pages = {109-125},
title = {On geometry of the set of admissible quadratic estimators of quadratic functions of normal parameters},
url = {http://eudml.org/doc/277042},
volume = {26},
year = {2006},
}

TY - JOUR
AU - Konrad Neumann
AU - Stefan Zontek
TI - On geometry of the set of admissible quadratic estimators of quadratic functions of normal parameters
JO - Discussiones Mathematicae Probability and Statistics
PY - 2006
VL - 26
IS - 2
SP - 109
EP - 125
AB - We consider the problem of admissible quadratic estimation of a linear function of μ² and σ² in n dimensional normal model N(Kμ,σ²Iₙ) under quadratic risk function. After reducing this problem to admissible estimation of a linear function of two quadratic forms, the set of admissible estimators are characterized by giving formulae on the boundary of the set D ⊂ R² of components of the two quadratic forms constituting the set of admissible estimators. Different shapes and topological properties of the set D are studied.
LA - eng
KW - linear estimator; quadratic estimator; Bayesian quadratic estimator; quadratic loss function; admissibility; quadratic subspace
UR - http://eudml.org/doc/277042
ER -

References

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  2. [2] S. Gnot and J. Kleffe, Quadratic estimation in mixed linear models with two variance components, J. Statist. Plann. Inference 8 (1983), 267-279. Zbl0561.62064
  3. [3] D.A. Harville, Quadratic unbiased estimation of two variance components for the one-way classification, Biometrika 56 (1969), 313-326. Zbl0184.22305
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  6. [6] K. Neumann and S. Zontek, On geometry of the set of admissible invariant quadratic estimators in balanced two variance components model, Statistical Papers 45 (2004), 67-80. Zbl1052.62072
  7. [7] A.L. Rukhin, Quadratic estimators of quadratic functions of normal parameters, J. Statist. Plann. Inference 15 (1987), 301-310. Zbl0609.62039
  8. [8] A.L. Rukhin, Admissible polynomial estimates for quadratic polynomials of normal parameters (in russian), Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. 184, Issued. Mat. Statist. 9 (1990), 234-247. Zbl0759.62002
  9. [9] R. Zmyślony, Quadratic admissible estimators, (in polish) Roczniki Polskiego Towarzystwa Matematycznego, Seria III: Matematyka Stosowana VII, (1976), 117-122. 
  10. [10] S. Zontek, Admissibility of limits of the unique locally best linear estimators with application to variance components models, Probab. Math. Statist. 9. 2 (1988), 29-44. Zbl0674.62008

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