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

top
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

top

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

top
  1. [1] S. Gnot, E. Rafajłowicz and A. Urbańska-Motyka, Statistical inference in a linear model for spatially located sensors and random input, Ann. Inst. Statist. Math. 53. 2 (2001), 370-379. Zbl1027.62038
  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
  4. [4] L.R. LaMotte, Admissibility in linear model, Ann. Statist. 19 (1982), 245-256. Zbl0485.62070
  5. [5] L.R. LaMotte, Admissibility, unbiasedness, and nonnegativity in the balanced, random, one-way anova model, Linear statistical inference (Poznań, 1984), Lecture Notes in Statist. 35 (1985), 184-199. 
  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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.